Research ArticleGlobal Bifurcation of Positive Solutions of AsymptoticallyLinear Elliptic Problems

Ruyun Ma Yanqiong Lu and Ruipeng Chen

Department of Mathematics Northwest Normal University Lanzhou 730070 China

Correspondence should be addressed to Ruyun Ma ruyun ma126com

Received 13 March 2014 Revised 25 April 2014 Accepted 8 May 2014 Published 25 May 2014

Academic Editor Gennaro Infante

Copyright copy 2014 Ruyun Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We are concerned with determining values of 120582 for which there exist positive solutions of the nonlinear elliptic problem minusΔ119906 =

120582119886(119909)119891(119906) in Ω 120597119906120597n + 119887(119909)119892(119906) = 0 on 120597ΩThe proof of our main results is based upon unilateral global bifurcation theoremof Lopez-Gomez

1 Introduction

Let Ω be a bounded domain of Euclidean space R119873 119873 ge 2with smooth boundary 120597Ω In this paper we consider thenonlinear elliptic boundary value problem

minusΔ119906 = 120582119886 (119909) 119891 (119906) inΩ

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(1)

where Δ = sum119873

119895=1(12059721205971199092

119895) 120582 gt 0 is a parameter and n is the

unit exterior normal to 120597ΩWe make the following assumptions

(H0) 119886 isin 119862120579(Ω) with 119886(119909) gt 0 in 119909 isin Ω 119887 isin 1198621+120579(120597Ω) with119887 ge 0 and 119887 equiv 0 on 120597Ω

(H1) 119891 isin 1198621(RR) is an odd function with 119891(119904) gt 0 for

119904 gt 0 and there exist constants 1198910 119891infinisin (0infin) and

functions 120585 ℎ isin 1198621(RR) such that

119891 (119904) = 1198910119904 + 120585 (119904) 120585 (119904) = 119900 (|119904|) as 119904 997888rarr 0 (2)

119891 (119904) = 119891infin119904 + ℎ (119904) ℎ (119904) = 119900 (|119904|) as 119904 997888rarr +infin (3)

(H2) 119892 isin 1198621(RR) is an odd function with 119892(119904) gt 0 for

119904 gt 0 and there exist constants 1198920 119892infinisin (0infin) and

functions 120577 119896 isin 1198621(RR) such that

119892 (119904) = 1198920119904 + 120577 (119904) 120577 (119904) = 119900 (|119904|) as 119904 997888rarr 0

119892 (119904) = 119892infin119904 + 119896 (119904) 119896 (119904) = 119900 (|119904|) as 119904 997888rarr +infin

(4)

A solution 119906 isin 1198622(Ω) of (1) is said to be positive if

119906 gt 0 on Ω The purpose of this paper is to study the globalbifurcation of positive solutions for the asymptotically linearelliptic eigenvalue problems (1)

Let119883 = 119862(Ω) be the space of continuous functions onΩThen it is a Banach space with the norm

119906 = max |119906 (119909)| | 119909 isin Ω (5)

Let

119875 = 119906 isin 119883 | 119906 (119909) ge 0 119909 isin Ω (6)

Then119875 is a conewhich is normal and has a nonempty interiorand119883 = 119875 minus 119875 Moreover

int119875 = 119906 isin 119875 119906 (119909) gt 0 for 119909 isin Ω (7)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 749368 7 pageshttpdxdoiorg1011552014749368

2 Abstract and Applied Analysis

By a constant 120582infin1

we denote the first eigenvalue of theeigenvalue problem

minusΔ120593 = 120582119886 (119909) 119891infin120593 in Ω

120597120593

120597n+ 119887 (119909) 119892infin120593 = 0 on 120597Ω

(8)

By a constant 12058201we denote the first eigenvalue of the

eigenvalue problem

minusΔ120593 = 120582119886 (119909) 1198910120593 in Ω

120597120593

120597n+ 119887 (119909) 1198920120593 = 0 on 120597Ω

(9)

It is well known (cf Krasnoselrsquoskii [1]) that for ] isin 0infin 120582]1

is positive and simple and that it is a unique eigenvalue withpositive eigenfunctions 120593]

1isin 1198622+120579(Ω) In what follows the

positive eigenfunction 120593]1is normalized as 120593]

1 = 1

Let S be the closure of the set (120582 119906) isin (0infin) times 119862(Ω) |

(120582 119906) is a positive solution of (1) in R times 119862(Ω)

Theorem 1 Let (H0)ndash(H2) hold Then there exists anunbounded closed and connected component C+ sub (0infin) times

119862(Ω) inS which joins (12058201 0) with (120582infin

1infin)

Corollary 2 Let (H0)ndash(H2) hold Assume that either

1205820

1lt 120582 lt 120582

infin

1 (10)

or

120582infin

1lt 120582 lt 120582

0

1 (11)

Then (1) has at least one positive solution

Remark 3 Ambrosetti et al [2] andUmezu [3 4] only studiedthe bifurcation from infinity for nonlinear elliptic eigenvalueproblems Nonlinear eigenvalue problems of ordinary dif-ferential equations have been extensively studied by manyauthors via fixed point theorem in cones and bifurcationstechniques see Henderson and Wang [5] and Ma [6 7] andthe references therein Ma andThompson [7] considered thetwo-point boundary value problem

11990610158401015840(119905) + 119903119886 (119905) 119891 (119906) = 0 119906 (0) = 119906 (1) = 0 (12)

By using the well-known Rabinowitz global bifurcation the-orem [8] they proved the following

Theorem A (see [7 Theorem 11]) Assume that

(A1) 119886 [0 1] rarr [0infin) is continuous and 119886(119905) equiv 0 onany subinterval of [0 1]

(A2) 119891 isin 119862(RR) with 119904119891(119904) gt 0 for 119904 = 0

(A3) there exist 1198910 119891infin

isin (0infin) such that 1198910

=

lim|119904|rarr0

(119891(119904)119904) and 119891infin= lim|119904|rarrinfin

(119891(119904)119904)

(A4) 119896is the 119896th-eigenvalue of 11990610158401015840(119905) + 120582119886(119905)119906 = 0 119906(0) =

119906(1) = 0

If either 119896119891infinlt 119903 lt

1198961198910or 1198961198910lt 119903 lt

119896119891infin then (12)

has two solutions 119906+119896and 119906minus

119896such that 119906+

119896has exactly 119896minus1 zero

in (0 1) and is positive near 0 and 119906minus119896has exactly 119896 minus 1 zero in

(0 1) and is negative near 0

Obviously Corollary 2 is a higher dimensional analogueof Ma andThompson [7 Theorem 11] with 119896 = 1

Remark 4 Shi [9] studied the exact number of all nontrivialsolutions for

minusΔ119906 = 120582119891 (119906) in Ω

119906 = 0 on 120597Ω(13)

for 120582 in certain parameter range He proved the existenceof global smooth branches of positive solutions by using theimplicit function theorem under some further restrictions on119891

Remark 5 Nonlinear elliptic eigenvalue problems have beenstudied in [4 10] via topological degree and global bifurcationtechniques The positone case 119891(0) equiv 0 is considered in [10]which is extended to the semipositone case 119891(0) lt 0 in [4]An emphasis is inTheorem 1 andCorollary 2 no assumptionimposed on the boundedness of the function ℎ in (3)

Remark 6 Precup [11] applied the Moser-Harnack inequal-ity for nonnegative superharmonic functions to produce asuitable cone and developed fixed point theorem in cones ofKrasnoselskii-type to discuss the existence andmultiplicity ofpositive solutions to elliptic boundary value problems

Δ119906 + 119891 (119906) = 0 in Ω

119906 (119909) gt 0 in Ω

119906 = 0 on 120597Ω

(14)

The constant119860 in [11 (31)] and the constant119861 in [11 (32)] arenot optimal so that [11 Theorem 31] is not sharp However(10) and (11) in Corollary 2 are optimal In fact for thefunction

119891lowast(119904) = 120582

0

1119904 + arctan( 119904

2

1 + 1199042) (15)

which satisfies 119891lowast0= 119891lowast

infin= 1205820

1 the elliptic problem

Δ119906 + 119891lowast(119906) = 0 in Ω

120597119906

120597n+ 119906 = 0 on 120597Ω

(16)

has no positive solution

The rest of this paper is organized as follows The proofof our main results is based upon the unilateral globalbifurcation theoremof Lopez-Gomez which is different fromthe topological degree arguments used in [2ndash4 10] So inSection 2 we state a preliminary result based upon unilateralglobal bifurcation theorem of Lopez-Gomez In Section 3we reduce (1) into a compact operator equation Section 4 isdevoted to the proof of Theorem 1

Abstract and Applied Analysis 3

2 Unilateral Global Bifurcation Theorem ofLoacutepez-Goacutemez

Let 119880 be a Banach space with the normal sdot Let L(119880)

stand for the space of linear continuous operators in 119880 Let119869 = (119886 119887) sub R Let F 119877 times 119880 rarr 119880 be a nonlinear operatorof the form

F (120582 119906) = 119868119880 minus 120582119870 +N (120582 119906) (17)

whereN 119869 times 119880 rarr 119880 is a continuous operator compact onbounded sets such that

N (120582 119906) = ∘ (119906) (18)

as 119906 rarr 0 uniformly in any compact interval of 119869 119870 isin

L(119880) is a linear compact operator and 1199030= 0 is a simple

characteristic value of119870 that is

ker [119868119880minus 1199030119870] = span [120593

0] (19)

for some 1205930isin 119880 0 satisfying

1205930notin 119877 [119868

119880minus 1199030119870] (20)

Let S be the closure of the set

(120582 119906) | (120582 119906) isin 119869 times 119880F (120582 119906) = 0 119906 = 0 (21)

Let C+ (resp Cminus) be the component of S that meets (1199030 0)

and around (1199030 0) lies in S 119876

minus

120598120578(resp S 119876

+

120598120578) see [12

Section 64] for the detailsLet

Σ = 120582 isin 119869 dim ker [119868119880minus 120582119870] ge 1 (22)

Then we present the unilateral global bifurcation theorem ofLopez-Gomez see [12 Theorem 643]

Lemma 7 (see [12] unilateral global bifurcation ofLopez-Gomez) Assume Σ is discrete 119903

0isin Σ satisfies

(19) and the index 119868119899119889(0N(120582)) changes sign as 120582 crosses 1199030

Then for ] isin + minus the component C] satisfies one of thefollowing

(i) C] is unbounded in R times 119880(ii) there exists 119903

1isin Σ 119903

0 such that (119903

1 0) isin C]

(iii) C] contains a point

(120582 119910) isin 119877 times (119884 0) (23)

where 119884 is the complement of ker[119868119880minus 1199030119870] in 119880

3 Reduction to a Compact Operator Equation

To establish Theorem 1 we begin with the reduction of (1)to a suitable equation for compact operators According toGilbarg and Trudinger [13] letK

infin 119862120579(Ω) rarr 119862

2+120579(Ω) be

the resolvent of the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 0 on 120597Ω

(24)

By Amann [14 Theorem 42] Kinfin

is uniquely extended toa linear mapping of 119862(Ω) compactly into 119862

1(Ω) and it is

strongly positive meaning that Kinfin120601 gt 0 on Ω for any

120601 isin 119862(Ω) with the condition that 120601 ge 0 and 120601 equiv 0 onΩLet R

infin 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 120595 on 120597Ω

(25)

According to Amann [15 Section 4] Rinfin

is uniquelyextended to a linear mapping of 119862(120597Ω) compactly into119862(Ω) By the standard regularity argument problem (1) isequivalent to the operator equation

119906 = 120582Kinfin[119886119891 (119906)] +R

infin [119887120591 (minus119896 (119906))] in 119862 (Ω) (26)

Here 120591 119862(Ω) rarr 119862(120597Ω) is the usual trace operatorSimilarly letK

0 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of

the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 0 on 120597Ω

(27)

Then K0is uniquely extended to a linear mapping of 119862(Ω)

compactly into 1198621(Ω) and it is strongly positiveLet R

0 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 120595 on 120597Ω

(28)

ThenR0is uniquely extended to a linear mapping of 119862(120597Ω)

compactly into 119862(Ω) Furthermore (1) is equivalent to theoperator equation

119906 = 120582K0[119886119891 (119906)] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω) (29)

4 The Proof of Main Results

Obviously (H1) and (H2) imply that

lim|119904|rarr0

120585 (119904)

|119904|= 0 lim

|119904|rarr0

120577 (119904)

|119904|= 0 (30)

lim|119904|rarrinfin

ℎ (119904)

|119904|= 0 lim

|119904|rarrinfin

119896 (119904)

|119904|= 0 (31)

Let

ℎ (119903) = max |ℎ (119904)| | 0 le 119904 le 119903

(119903) = max |119896 (119904)| | 0 le 119904 le 119903 (32)

4 Abstract and Applied Analysis

Then ℎ and are nondecreasing and there exist 119903119899 infin such

that

lim119899rarrinfin

ℎ (119903119899)

119903119899

= lim119899rarrinfin

(119903119899)

119903119899

= 0 (33)

Indeed for any 119903 gt 0 there exists 119904 le 119903 such that ℎ(119903)119903 =|ℎ(119904)|119903 Additionally if we assume 119904 le 119862 for some 119862 gt 0then it follows that ℎ(119903)119903 le 119862119903 rarr 0 as 119903 rarr infin On theother hand we assume that there exists 119903

119899such that 119904

119899rarr infin

then it follows from (31) that ℎ(119903119899)119903119899le |ℎ(119904

119899)|119904119899rarr 0 as

119895 rarr infin as desiredWe consider

119906 = 120582K0[119886 (1198910119906 + 120585 (119906))] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(34)

as a bifurcation problem from the trivial solution 119906 equiv 0Define119870 119883 rarr 119883

119870119906 (119905) = K0[1198861198910119906] (119905) (35)

then 119870 is a strongly positive linear operator on 119883 It is easyto verify that119870 119875 rarr 119875 is completely continuous From [14Theorem 32] it follows that

119903 (119870) = [1205820

1]minus1

(36)

DefineN [0infin) times 119883 rarr 119883 by

N (120582 119906) = 120582K0 [119886120585 (119906)] +R0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(37)

then we have from (30) that

N (120582 119906) = ∘ (119906) (38)

locally and uniformly in 120582It is very easy to check that (34) enjoys the structural

requirements for applying the unilateral global bifurcationtheory of [12 Sections 64 65] (by a counter example ofDancer [16] the global unilateral theorem of Rabinowitz [8]is false as stated So it cannot be used) As the theoremof Crandall and Rabinowitz [17] is applied to get the localbifurcation to positive solutions from (120582

0

1 0) the algebraic

multiplicity of Esquinas and Lopez-Gomez [18] (see [19Chapter 4]) equals 1 and therefore by [12 Theorem 562]or [19 Proposition 1231] the local index of 0 as a fixedpoint of 119868 minus 120582K

0changes sign as 120582 crosses 1205820

1 Therefore it

follows from Lemma 7 that there exists the component C+that satisfies one of the following

(i) C+ is unbounded in R times 119880

(ii) there exists 120582lowast isin (0infin) with 120582lowast = 1205820

1and 120595 isin 119862(Ω)

which changes its sign onΩ such that

[119868 minus 120582lowastK0] 120595 = 0 (120582

lowast 0) isin C

+ (39)

(iii) C+ contains a point

(120582 119910) isin 119877 times (119884 0) (40)

where 119884 is the complement of ker[119868119883minus 1205820

1K0(119886(sdot))]

in119883

In what follows we will show that the above Case (ii) andCase (iii) do not occur

In fact if (120583 119910) isin C+ is a nontrivial solution of (34) then119910 satisfies the problem

minusΔ119910 = 120583119886 (119909) 119891 (119910) in Ω

120597119910

120597n+ 119887 (119909) 119892 (119910) = 0 on 120597Ω

(41)

that is to say 119910 satisfies the linear problem

minusΔ119910 = 120583119886 (119909) 119891 (119909) 119910 in Ω

120597119910

120597n+ 119887 (119909) 119892 (119909) 119910 = 0 on 120597Ω

(42)

where

119892 (119909) =

119892 (119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198920 as 119910 (119909) = 0

119891 (119909) =

119891(119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198910 as 119910 (119909) = 0

(43)

We claim that

119910 isin C+ 119910 = 0 997904rArr 119910 isin int119875 cup int (minus119875) (44)

Suppose on the contrary that (120583 119910) isin C+ with 119910 = 0 and119910 isin 120597119875Then there exists a sequence (120583

119899 119906119899) sub C+cap119875with

119906119899gt 0 for 119899 isin N such that

(120583119899 119906119899) 997888rarr (120583 119910) as 119899 997888rarr infin (45)

and consequently

119910 (119909) ge 0 119909 isin Ω (46)

Combining this with the fact that (120583 119910) is a nontrivialsolution of (34) and using the strongmaximumprinciple [20Theorem 24] and (42) it concludes that

119910 (119909) gt 0 119909 isin Ω (47)

contracting 119910 isin 120597119875Similarly for (120578 119910) isin C+ with 119910 = 0 and 119910 isin 120597(minus119875) we

get the desired contradictionTherefore the claim (44) is trueSince 119891 and 119892 are odd in R (120582 minus119906) is a solution of (34)

if and only if (120582 119906) is a solution of (34) Combining this and(44) and using the fact that the eigenfunctions correspondingto the eigenvalue 120582

119895of the operator 119870 with 119895 ge 2 have

Abstract and Applied Analysis 5

to change its sign in Ω it concludes that Case (ii) andCase (iii) cannot occurTherefore there exists an unboundedconnected subset C+ of the set

(120582 119906) isin (0infin) times 119875 119906 = 119860 (120582 119906) 119906 isin int119875 cup (12058201 0)

(48)

such that (12058201 0) isin C+

Proof of Theorem 1 It is clear that any solution of (34) of theform (120582 119906) yields a solution 119906 of (1) We will show that C+joins (1205820

1 0) to (120582infin

1infin)

Let (120583119899 119910119899) isin C+ satisfy

120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817 997888rarr infin (49)

We note that 120583119899gt 0 for all 119899 isin N since 119906 = 0 is the only

solution of (34) (ie (26)) for 120582 = 0In fact suppose on the contrary that 119906 is a nontrivial

solution of the problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(50)

then 119906 satisfies the linear problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ (119909) 119906 = 0 on 120597Ω

(51)

where (119909) = 119887(119909)(119892(119906)119906) Together with (H2) and theresults of Krasnoselrsquoskii [1] it follows that 119906 equiv 0 in Ω whichis a contradiction Therefore (34) (ie (26)) with 120582 = 0 hasonly trivial solution

Case 1 (120582infin1lt 120582 lt 120582

0

1) In this case we show that

(120582infin

1 1205820

1) sube (120582 isin R | exist (120582 119906) isin C

+) (52)

We divide the proof into two steps

Step 1We show that if there exists a constant number119872 gt 0

such that

120583119899sub (0119872] (53)

then C+ joins (12058201 0) to (120582infin

1infin)

From (53) we have 119910119899 rarr infin We divide the equation

119910119899= 120583119899Kinfin[119886 (119891infin119910119899+ ℎ (119910

119899))] +R

infin[119887120591 (minus119896 (119910

119899))]

in 119862 (Ω) (54)

by 119910119899 and set V

119899= 119910119899119910119899 Since V

119899is bounded in 119883

choosing a subsequence and relabeling if necessary we seethat V

119899rarr V for some V isin 119883 with V = 1 Moreover

from (31) and the fact that ℎ and are nondecreasing togetherwith the assertion 119910

119899 rarr infin there exists some 119903

119899119895le 119910119899

such that ℎ(119903119899119895) ge ℎ119910

119899 and 119903

119899119895rarr infin It follows that

ℎ(119910119899)119910119899 le ℎ(119903

119899119895)119903119899119895

rarr 0 as 119899 rarr infin Subsequentlywe have

lim119899rarrinfin

1003816100381610038161003816ℎ (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= lim119899rarrinfin

1003816100381610038161003816119896 (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= 0 (55)

since |ℎ(119910119899(119905))|119910

119899 le ℎ(|119910

119899(119905)|)119910

119899 le ℎ(119910

119899)119910119899 and

|119896(119910119899(119905))|119910

119899 le (|119910

119899(119905)|)119910

119899 le (119910

119899)119910119899 Therefore

V = 120583Kinfin[119886119891infinV] in 119862 (Ω) (56)

where 120583 = lim119899rarrinfin

120583119899 again choosing a subsequence and

relabeling if necessaryThus

minusΔV = 120583119886 (119909) 119891infinV in Ω

120597V120597n

+ 119887 (119909) 119892infinV = 0 on 120597Ω(57)

Since V = 1 and V ge 0 the strong positivity ofKinfin

ensuresthat

V gt 0 on Ω (58)

Thus 120583 = 120582infin1 and accordinglyC joins (1205820

1 0) to (120582infin

1infin)

Step 2We show that there exists a constant119872 such that 120583119899isin

(0119872] for all 119899From (H1) and (H2) there exist constants 120572 120573 isin (0infin)

such that

119891 (119906) ge 120572119906 119906 isin [0infin)

119892 (119906) le 120573119906 119906 isin [0infin)

(59)

By the same method to defineKinfinandR

infinin Section 3

we may defineKlowast andRlowast as followsLetKlowast 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear

boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(60)

Then Klowast is uniquely extended to a linear mapping of 119862(Ω)compactly into 1198621(Ω) and it is strongly positive Let Rlowast 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear boundary

value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 120595 on 120597Ω

(61)

Let 120582lowast1be the eigenvalue of the linear problem

minusΔ119906 = 120582119886 (119909) 120572119906 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(62)

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

By a constant 120582infin1

we denote the first eigenvalue of theeigenvalue problem

minusΔ120593 = 120582119886 (119909) 119891infin120593 in Ω

120597120593

120597n+ 119887 (119909) 119892infin120593 = 0 on 120597Ω

(8)

By a constant 12058201we denote the first eigenvalue of the

eigenvalue problem

minusΔ120593 = 120582119886 (119909) 1198910120593 in Ω

120597120593

120597n+ 119887 (119909) 1198920120593 = 0 on 120597Ω

(9)

It is well known (cf Krasnoselrsquoskii [1]) that for ] isin 0infin 120582]1

is positive and simple and that it is a unique eigenvalue withpositive eigenfunctions 120593]

1isin 1198622+120579(Ω) In what follows the

positive eigenfunction 120593]1is normalized as 120593]

1 = 1

Let S be the closure of the set (120582 119906) isin (0infin) times 119862(Ω) |

(120582 119906) is a positive solution of (1) in R times 119862(Ω)

Theorem 1 Let (H0)ndash(H2) hold Then there exists anunbounded closed and connected component C+ sub (0infin) times

119862(Ω) inS which joins (12058201 0) with (120582infin

1infin)

Corollary 2 Let (H0)ndash(H2) hold Assume that either

1205820

1lt 120582 lt 120582

infin

1 (10)

or

120582infin

1lt 120582 lt 120582

0

1 (11)

Then (1) has at least one positive solution

Remark 3 Ambrosetti et al [2] andUmezu [3 4] only studiedthe bifurcation from infinity for nonlinear elliptic eigenvalueproblems Nonlinear eigenvalue problems of ordinary dif-ferential equations have been extensively studied by manyauthors via fixed point theorem in cones and bifurcationstechniques see Henderson and Wang [5] and Ma [6 7] andthe references therein Ma andThompson [7] considered thetwo-point boundary value problem

11990610158401015840(119905) + 119903119886 (119905) 119891 (119906) = 0 119906 (0) = 119906 (1) = 0 (12)

By using the well-known Rabinowitz global bifurcation the-orem [8] they proved the following

Theorem A (see [7 Theorem 11]) Assume that

(A1) 119886 [0 1] rarr [0infin) is continuous and 119886(119905) equiv 0 onany subinterval of [0 1]

(A2) 119891 isin 119862(RR) with 119904119891(119904) gt 0 for 119904 = 0

(A3) there exist 1198910 119891infin

isin (0infin) such that 1198910

=

lim|119904|rarr0

(119891(119904)119904) and 119891infin= lim|119904|rarrinfin

(119891(119904)119904)

(A4) 119896is the 119896th-eigenvalue of 11990610158401015840(119905) + 120582119886(119905)119906 = 0 119906(0) =

119906(1) = 0

If either 119896119891infinlt 119903 lt

1198961198910or 1198961198910lt 119903 lt

119896119891infin then (12)

has two solutions 119906+119896and 119906minus

119896such that 119906+

119896has exactly 119896minus1 zero

in (0 1) and is positive near 0 and 119906minus119896has exactly 119896 minus 1 zero in

(0 1) and is negative near 0

Obviously Corollary 2 is a higher dimensional analogueof Ma andThompson [7 Theorem 11] with 119896 = 1

Remark 4 Shi [9] studied the exact number of all nontrivialsolutions for

minusΔ119906 = 120582119891 (119906) in Ω

119906 = 0 on 120597Ω(13)

for 120582 in certain parameter range He proved the existenceof global smooth branches of positive solutions by using theimplicit function theorem under some further restrictions on119891

Remark 5 Nonlinear elliptic eigenvalue problems have beenstudied in [4 10] via topological degree and global bifurcationtechniques The positone case 119891(0) equiv 0 is considered in [10]which is extended to the semipositone case 119891(0) lt 0 in [4]An emphasis is inTheorem 1 andCorollary 2 no assumptionimposed on the boundedness of the function ℎ in (3)

Remark 6 Precup [11] applied the Moser-Harnack inequal-ity for nonnegative superharmonic functions to produce asuitable cone and developed fixed point theorem in cones ofKrasnoselskii-type to discuss the existence andmultiplicity ofpositive solutions to elliptic boundary value problems

Δ119906 + 119891 (119906) = 0 in Ω

119906 (119909) gt 0 in Ω

119906 = 0 on 120597Ω

(14)

The constant119860 in [11 (31)] and the constant119861 in [11 (32)] arenot optimal so that [11 Theorem 31] is not sharp However(10) and (11) in Corollary 2 are optimal In fact for thefunction

119891lowast(119904) = 120582

0

1119904 + arctan( 119904

2

1 + 1199042) (15)

which satisfies 119891lowast0= 119891lowast

infin= 1205820

1 the elliptic problem

Δ119906 + 119891lowast(119906) = 0 in Ω

120597119906

120597n+ 119906 = 0 on 120597Ω

(16)

has no positive solution

The rest of this paper is organized as follows The proofof our main results is based upon the unilateral globalbifurcation theoremof Lopez-Gomez which is different fromthe topological degree arguments used in [2ndash4 10] So inSection 2 we state a preliminary result based upon unilateralglobal bifurcation theorem of Lopez-Gomez In Section 3we reduce (1) into a compact operator equation Section 4 isdevoted to the proof of Theorem 1

Abstract and Applied Analysis 3

2 Unilateral Global Bifurcation Theorem ofLoacutepez-Goacutemez

Let 119880 be a Banach space with the normal sdot Let L(119880)

stand for the space of linear continuous operators in 119880 Let119869 = (119886 119887) sub R Let F 119877 times 119880 rarr 119880 be a nonlinear operatorof the form

F (120582 119906) = 119868119880 minus 120582119870 +N (120582 119906) (17)

whereN 119869 times 119880 rarr 119880 is a continuous operator compact onbounded sets such that

N (120582 119906) = ∘ (119906) (18)

as 119906 rarr 0 uniformly in any compact interval of 119869 119870 isin

L(119880) is a linear compact operator and 1199030= 0 is a simple

characteristic value of119870 that is

ker [119868119880minus 1199030119870] = span [120593

0] (19)

for some 1205930isin 119880 0 satisfying

1205930notin 119877 [119868

119880minus 1199030119870] (20)

Let S be the closure of the set

(120582 119906) | (120582 119906) isin 119869 times 119880F (120582 119906) = 0 119906 = 0 (21)

Let C+ (resp Cminus) be the component of S that meets (1199030 0)

and around (1199030 0) lies in S 119876

minus

120598120578(resp S 119876

+

120598120578) see [12

Section 64] for the detailsLet

Σ = 120582 isin 119869 dim ker [119868119880minus 120582119870] ge 1 (22)

Then we present the unilateral global bifurcation theorem ofLopez-Gomez see [12 Theorem 643]

Lemma 7 (see [12] unilateral global bifurcation ofLopez-Gomez) Assume Σ is discrete 119903

0isin Σ satisfies

(19) and the index 119868119899119889(0N(120582)) changes sign as 120582 crosses 1199030

Then for ] isin + minus the component C] satisfies one of thefollowing

(i) C] is unbounded in R times 119880(ii) there exists 119903

1isin Σ 119903

0 such that (119903

1 0) isin C]

(iii) C] contains a point

(120582 119910) isin 119877 times (119884 0) (23)

where 119884 is the complement of ker[119868119880minus 1199030119870] in 119880

3 Reduction to a Compact Operator Equation

To establish Theorem 1 we begin with the reduction of (1)to a suitable equation for compact operators According toGilbarg and Trudinger [13] letK

infin 119862120579(Ω) rarr 119862

2+120579(Ω) be

the resolvent of the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 0 on 120597Ω

(24)

By Amann [14 Theorem 42] Kinfin

is uniquely extended toa linear mapping of 119862(Ω) compactly into 119862

1(Ω) and it is

strongly positive meaning that Kinfin120601 gt 0 on Ω for any

120601 isin 119862(Ω) with the condition that 120601 ge 0 and 120601 equiv 0 onΩLet R

infin 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 120595 on 120597Ω

(25)

According to Amann [15 Section 4] Rinfin

is uniquelyextended to a linear mapping of 119862(120597Ω) compactly into119862(Ω) By the standard regularity argument problem (1) isequivalent to the operator equation

119906 = 120582Kinfin[119886119891 (119906)] +R

infin [119887120591 (minus119896 (119906))] in 119862 (Ω) (26)

Here 120591 119862(Ω) rarr 119862(120597Ω) is the usual trace operatorSimilarly letK

0 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of

the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 0 on 120597Ω

(27)

Then K0is uniquely extended to a linear mapping of 119862(Ω)

compactly into 1198621(Ω) and it is strongly positiveLet R

0 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 120595 on 120597Ω

(28)

ThenR0is uniquely extended to a linear mapping of 119862(120597Ω)

compactly into 119862(Ω) Furthermore (1) is equivalent to theoperator equation

119906 = 120582K0[119886119891 (119906)] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω) (29)

4 The Proof of Main Results

Obviously (H1) and (H2) imply that

lim|119904|rarr0

120585 (119904)

|119904|= 0 lim

|119904|rarr0

120577 (119904)

|119904|= 0 (30)

lim|119904|rarrinfin

ℎ (119904)

|119904|= 0 lim

|119904|rarrinfin

119896 (119904)

|119904|= 0 (31)

Let

ℎ (119903) = max |ℎ (119904)| | 0 le 119904 le 119903

(119903) = max |119896 (119904)| | 0 le 119904 le 119903 (32)

4 Abstract and Applied Analysis

Then ℎ and are nondecreasing and there exist 119903119899 infin such

that

lim119899rarrinfin

ℎ (119903119899)

119903119899

= lim119899rarrinfin

(119903119899)

119903119899

= 0 (33)

Indeed for any 119903 gt 0 there exists 119904 le 119903 such that ℎ(119903)119903 =|ℎ(119904)|119903 Additionally if we assume 119904 le 119862 for some 119862 gt 0then it follows that ℎ(119903)119903 le 119862119903 rarr 0 as 119903 rarr infin On theother hand we assume that there exists 119903

119899such that 119904

119899rarr infin

then it follows from (31) that ℎ(119903119899)119903119899le |ℎ(119904

119899)|119904119899rarr 0 as

119895 rarr infin as desiredWe consider

119906 = 120582K0[119886 (1198910119906 + 120585 (119906))] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(34)

as a bifurcation problem from the trivial solution 119906 equiv 0Define119870 119883 rarr 119883

119870119906 (119905) = K0[1198861198910119906] (119905) (35)

then 119870 is a strongly positive linear operator on 119883 It is easyto verify that119870 119875 rarr 119875 is completely continuous From [14Theorem 32] it follows that

119903 (119870) = [1205820

1]minus1

(36)

DefineN [0infin) times 119883 rarr 119883 by

N (120582 119906) = 120582K0 [119886120585 (119906)] +R0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(37)

then we have from (30) that

N (120582 119906) = ∘ (119906) (38)

locally and uniformly in 120582It is very easy to check that (34) enjoys the structural

requirements for applying the unilateral global bifurcationtheory of [12 Sections 64 65] (by a counter example ofDancer [16] the global unilateral theorem of Rabinowitz [8]is false as stated So it cannot be used) As the theoremof Crandall and Rabinowitz [17] is applied to get the localbifurcation to positive solutions from (120582

0

1 0) the algebraic

multiplicity of Esquinas and Lopez-Gomez [18] (see [19Chapter 4]) equals 1 and therefore by [12 Theorem 562]or [19 Proposition 1231] the local index of 0 as a fixedpoint of 119868 minus 120582K

0changes sign as 120582 crosses 1205820

1 Therefore it

follows from Lemma 7 that there exists the component C+that satisfies one of the following

(i) C+ is unbounded in R times 119880

(ii) there exists 120582lowast isin (0infin) with 120582lowast = 1205820

1and 120595 isin 119862(Ω)

which changes its sign onΩ such that

[119868 minus 120582lowastK0] 120595 = 0 (120582

lowast 0) isin C

+ (39)

(iii) C+ contains a point

(120582 119910) isin 119877 times (119884 0) (40)

where 119884 is the complement of ker[119868119883minus 1205820

1K0(119886(sdot))]

in119883

In what follows we will show that the above Case (ii) andCase (iii) do not occur

In fact if (120583 119910) isin C+ is a nontrivial solution of (34) then119910 satisfies the problem

minusΔ119910 = 120583119886 (119909) 119891 (119910) in Ω

120597119910

120597n+ 119887 (119909) 119892 (119910) = 0 on 120597Ω

(41)

that is to say 119910 satisfies the linear problem

minusΔ119910 = 120583119886 (119909) 119891 (119909) 119910 in Ω

120597119910

120597n+ 119887 (119909) 119892 (119909) 119910 = 0 on 120597Ω

(42)

where

119892 (119909) =

119892 (119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198920 as 119910 (119909) = 0

119891 (119909) =

119891(119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198910 as 119910 (119909) = 0

(43)

We claim that

119910 isin C+ 119910 = 0 997904rArr 119910 isin int119875 cup int (minus119875) (44)

Suppose on the contrary that (120583 119910) isin C+ with 119910 = 0 and119910 isin 120597119875Then there exists a sequence (120583

119899 119906119899) sub C+cap119875with

119906119899gt 0 for 119899 isin N such that

(120583119899 119906119899) 997888rarr (120583 119910) as 119899 997888rarr infin (45)

and consequently

119910 (119909) ge 0 119909 isin Ω (46)

Combining this with the fact that (120583 119910) is a nontrivialsolution of (34) and using the strongmaximumprinciple [20Theorem 24] and (42) it concludes that

119910 (119909) gt 0 119909 isin Ω (47)

contracting 119910 isin 120597119875Similarly for (120578 119910) isin C+ with 119910 = 0 and 119910 isin 120597(minus119875) we

get the desired contradictionTherefore the claim (44) is trueSince 119891 and 119892 are odd in R (120582 minus119906) is a solution of (34)

if and only if (120582 119906) is a solution of (34) Combining this and(44) and using the fact that the eigenfunctions correspondingto the eigenvalue 120582

119895of the operator 119870 with 119895 ge 2 have

Abstract and Applied Analysis 5

to change its sign in Ω it concludes that Case (ii) andCase (iii) cannot occurTherefore there exists an unboundedconnected subset C+ of the set

(120582 119906) isin (0infin) times 119875 119906 = 119860 (120582 119906) 119906 isin int119875 cup (12058201 0)

(48)

such that (12058201 0) isin C+

Proof of Theorem 1 It is clear that any solution of (34) of theform (120582 119906) yields a solution 119906 of (1) We will show that C+joins (1205820

1 0) to (120582infin

1infin)

Let (120583119899 119910119899) isin C+ satisfy

120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817 997888rarr infin (49)

We note that 120583119899gt 0 for all 119899 isin N since 119906 = 0 is the only

solution of (34) (ie (26)) for 120582 = 0In fact suppose on the contrary that 119906 is a nontrivial

solution of the problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(50)

then 119906 satisfies the linear problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ (119909) 119906 = 0 on 120597Ω

(51)

where (119909) = 119887(119909)(119892(119906)119906) Together with (H2) and theresults of Krasnoselrsquoskii [1] it follows that 119906 equiv 0 in Ω whichis a contradiction Therefore (34) (ie (26)) with 120582 = 0 hasonly trivial solution

Case 1 (120582infin1lt 120582 lt 120582

0

1) In this case we show that

(120582infin

1 1205820

1) sube (120582 isin R | exist (120582 119906) isin C

+) (52)

We divide the proof into two steps

Step 1We show that if there exists a constant number119872 gt 0

such that

120583119899sub (0119872] (53)

then C+ joins (12058201 0) to (120582infin

1infin)

From (53) we have 119910119899 rarr infin We divide the equation

119910119899= 120583119899Kinfin[119886 (119891infin119910119899+ ℎ (119910

119899))] +R

infin[119887120591 (minus119896 (119910

119899))]

in 119862 (Ω) (54)

by 119910119899 and set V

119899= 119910119899119910119899 Since V

119899is bounded in 119883

choosing a subsequence and relabeling if necessary we seethat V

119899rarr V for some V isin 119883 with V = 1 Moreover

from (31) and the fact that ℎ and are nondecreasing togetherwith the assertion 119910

119899 rarr infin there exists some 119903

119899119895le 119910119899

such that ℎ(119903119899119895) ge ℎ119910

119899 and 119903

119899119895rarr infin It follows that

ℎ(119910119899)119910119899 le ℎ(119903

119899119895)119903119899119895

rarr 0 as 119899 rarr infin Subsequentlywe have

lim119899rarrinfin

1003816100381610038161003816ℎ (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= lim119899rarrinfin

1003816100381610038161003816119896 (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= 0 (55)

since |ℎ(119910119899(119905))|119910

119899 le ℎ(|119910

119899(119905)|)119910

119899 le ℎ(119910

119899)119910119899 and

|119896(119910119899(119905))|119910

119899 le (|119910

119899(119905)|)119910

119899 le (119910

119899)119910119899 Therefore

V = 120583Kinfin[119886119891infinV] in 119862 (Ω) (56)

where 120583 = lim119899rarrinfin

120583119899 again choosing a subsequence and

relabeling if necessaryThus

minusΔV = 120583119886 (119909) 119891infinV in Ω

120597V120597n

+ 119887 (119909) 119892infinV = 0 on 120597Ω(57)

Since V = 1 and V ge 0 the strong positivity ofKinfin

ensuresthat

V gt 0 on Ω (58)

Thus 120583 = 120582infin1 and accordinglyC joins (1205820

1 0) to (120582infin

1infin)

Step 2We show that there exists a constant119872 such that 120583119899isin

(0119872] for all 119899From (H1) and (H2) there exist constants 120572 120573 isin (0infin)

such that

119891 (119906) ge 120572119906 119906 isin [0infin)

119892 (119906) le 120573119906 119906 isin [0infin)

(59)

By the same method to defineKinfinandR

infinin Section 3

we may defineKlowast andRlowast as followsLetKlowast 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear

boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(60)

Then Klowast is uniquely extended to a linear mapping of 119862(Ω)compactly into 1198621(Ω) and it is strongly positive Let Rlowast 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear boundary

value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 120595 on 120597Ω

(61)

Let 120582lowast1be the eigenvalue of the linear problem

minusΔ119906 = 120582119886 (119909) 120572119906 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(62)

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

2 Unilateral Global Bifurcation Theorem ofLoacutepez-Goacutemez

Let 119880 be a Banach space with the normal sdot Let L(119880)

stand for the space of linear continuous operators in 119880 Let119869 = (119886 119887) sub R Let F 119877 times 119880 rarr 119880 be a nonlinear operatorof the form

F (120582 119906) = 119868119880 minus 120582119870 +N (120582 119906) (17)

whereN 119869 times 119880 rarr 119880 is a continuous operator compact onbounded sets such that

N (120582 119906) = ∘ (119906) (18)

as 119906 rarr 0 uniformly in any compact interval of 119869 119870 isin

L(119880) is a linear compact operator and 1199030= 0 is a simple

characteristic value of119870 that is

ker [119868119880minus 1199030119870] = span [120593

0] (19)

for some 1205930isin 119880 0 satisfying

1205930notin 119877 [119868

119880minus 1199030119870] (20)

Let S be the closure of the set

(120582 119906) | (120582 119906) isin 119869 times 119880F (120582 119906) = 0 119906 = 0 (21)

Let C+ (resp Cminus) be the component of S that meets (1199030 0)

and around (1199030 0) lies in S 119876

minus

120598120578(resp S 119876

+

120598120578) see [12

Section 64] for the detailsLet

Σ = 120582 isin 119869 dim ker [119868119880minus 120582119870] ge 1 (22)

Then we present the unilateral global bifurcation theorem ofLopez-Gomez see [12 Theorem 643]

Lemma 7 (see [12] unilateral global bifurcation ofLopez-Gomez) Assume Σ is discrete 119903

0isin Σ satisfies

(19) and the index 119868119899119889(0N(120582)) changes sign as 120582 crosses 1199030

Then for ] isin + minus the component C] satisfies one of thefollowing

(i) C] is unbounded in R times 119880(ii) there exists 119903

1isin Σ 119903

0 such that (119903

1 0) isin C]

(iii) C] contains a point

(120582 119910) isin 119877 times (119884 0) (23)

where 119884 is the complement of ker[119868119880minus 1199030119870] in 119880

3 Reduction to a Compact Operator Equation

To establish Theorem 1 we begin with the reduction of (1)to a suitable equation for compact operators According toGilbarg and Trudinger [13] letK

infin 119862120579(Ω) rarr 119862

2+120579(Ω) be

the resolvent of the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 0 on 120597Ω

(24)

By Amann [14 Theorem 42] Kinfin

is uniquely extended toa linear mapping of 119862(Ω) compactly into 119862

1(Ω) and it is

strongly positive meaning that Kinfin120601 gt 0 on Ω for any

120601 isin 119862(Ω) with the condition that 120601 ge 0 and 120601 equiv 0 onΩLet R

infin 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892infin119906 = 120595 on 120597Ω

(25)

According to Amann [15 Section 4] Rinfin

is uniquelyextended to a linear mapping of 119862(120597Ω) compactly into119862(Ω) By the standard regularity argument problem (1) isequivalent to the operator equation

119906 = 120582Kinfin[119886119891 (119906)] +R

infin [119887120591 (minus119896 (119906))] in 119862 (Ω) (26)

Here 120591 119862(Ω) rarr 119862(120597Ω) is the usual trace operatorSimilarly letK

0 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of

the linear boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 0 on 120597Ω

(27)

Then K0is uniquely extended to a linear mapping of 119862(Ω)

compactly into 1198621(Ω) and it is strongly positiveLet R

0 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the

linear boundary value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 1198920119906 = 120595 on 120597Ω

(28)

ThenR0is uniquely extended to a linear mapping of 119862(120597Ω)

compactly into 119862(Ω) Furthermore (1) is equivalent to theoperator equation

119906 = 120582K0[119886119891 (119906)] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω) (29)

4 The Proof of Main Results

Obviously (H1) and (H2) imply that

lim|119904|rarr0

120585 (119904)

|119904|= 0 lim

|119904|rarr0

120577 (119904)

|119904|= 0 (30)

lim|119904|rarrinfin

ℎ (119904)

|119904|= 0 lim

|119904|rarrinfin

119896 (119904)

|119904|= 0 (31)

Let

ℎ (119903) = max |ℎ (119904)| | 0 le 119904 le 119903

(119903) = max |119896 (119904)| | 0 le 119904 le 119903 (32)

4 Abstract and Applied Analysis

Then ℎ and are nondecreasing and there exist 119903119899 infin such

that

lim119899rarrinfin

ℎ (119903119899)

119903119899

= lim119899rarrinfin

(119903119899)

119903119899

= 0 (33)

Indeed for any 119903 gt 0 there exists 119904 le 119903 such that ℎ(119903)119903 =|ℎ(119904)|119903 Additionally if we assume 119904 le 119862 for some 119862 gt 0then it follows that ℎ(119903)119903 le 119862119903 rarr 0 as 119903 rarr infin On theother hand we assume that there exists 119903

119899such that 119904

119899rarr infin

then it follows from (31) that ℎ(119903119899)119903119899le |ℎ(119904

119899)|119904119899rarr 0 as

119895 rarr infin as desiredWe consider

119906 = 120582K0[119886 (1198910119906 + 120585 (119906))] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(34)

as a bifurcation problem from the trivial solution 119906 equiv 0Define119870 119883 rarr 119883

119870119906 (119905) = K0[1198861198910119906] (119905) (35)

then 119870 is a strongly positive linear operator on 119883 It is easyto verify that119870 119875 rarr 119875 is completely continuous From [14Theorem 32] it follows that

119903 (119870) = [1205820

1]minus1

(36)

DefineN [0infin) times 119883 rarr 119883 by

N (120582 119906) = 120582K0 [119886120585 (119906)] +R0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(37)

then we have from (30) that

N (120582 119906) = ∘ (119906) (38)

locally and uniformly in 120582It is very easy to check that (34) enjoys the structural

requirements for applying the unilateral global bifurcationtheory of [12 Sections 64 65] (by a counter example ofDancer [16] the global unilateral theorem of Rabinowitz [8]is false as stated So it cannot be used) As the theoremof Crandall and Rabinowitz [17] is applied to get the localbifurcation to positive solutions from (120582

0

1 0) the algebraic

multiplicity of Esquinas and Lopez-Gomez [18] (see [19Chapter 4]) equals 1 and therefore by [12 Theorem 562]or [19 Proposition 1231] the local index of 0 as a fixedpoint of 119868 minus 120582K

0changes sign as 120582 crosses 1205820

1 Therefore it

follows from Lemma 7 that there exists the component C+that satisfies one of the following

(i) C+ is unbounded in R times 119880

(ii) there exists 120582lowast isin (0infin) with 120582lowast = 1205820

1and 120595 isin 119862(Ω)

which changes its sign onΩ such that

[119868 minus 120582lowastK0] 120595 = 0 (120582

lowast 0) isin C

+ (39)

(iii) C+ contains a point

(120582 119910) isin 119877 times (119884 0) (40)

where 119884 is the complement of ker[119868119883minus 1205820

1K0(119886(sdot))]

in119883

In what follows we will show that the above Case (ii) andCase (iii) do not occur

In fact if (120583 119910) isin C+ is a nontrivial solution of (34) then119910 satisfies the problem

minusΔ119910 = 120583119886 (119909) 119891 (119910) in Ω

120597119910

120597n+ 119887 (119909) 119892 (119910) = 0 on 120597Ω

(41)

that is to say 119910 satisfies the linear problem

minusΔ119910 = 120583119886 (119909) 119891 (119909) 119910 in Ω

120597119910

120597n+ 119887 (119909) 119892 (119909) 119910 = 0 on 120597Ω

(42)

where

119892 (119909) =

119892 (119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198920 as 119910 (119909) = 0

119891 (119909) =

119891(119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198910 as 119910 (119909) = 0

(43)

We claim that

119910 isin C+ 119910 = 0 997904rArr 119910 isin int119875 cup int (minus119875) (44)

Suppose on the contrary that (120583 119910) isin C+ with 119910 = 0 and119910 isin 120597119875Then there exists a sequence (120583

119899 119906119899) sub C+cap119875with

119906119899gt 0 for 119899 isin N such that

(120583119899 119906119899) 997888rarr (120583 119910) as 119899 997888rarr infin (45)

and consequently

119910 (119909) ge 0 119909 isin Ω (46)

Combining this with the fact that (120583 119910) is a nontrivialsolution of (34) and using the strongmaximumprinciple [20Theorem 24] and (42) it concludes that

119910 (119909) gt 0 119909 isin Ω (47)

contracting 119910 isin 120597119875Similarly for (120578 119910) isin C+ with 119910 = 0 and 119910 isin 120597(minus119875) we

get the desired contradictionTherefore the claim (44) is trueSince 119891 and 119892 are odd in R (120582 minus119906) is a solution of (34)

if and only if (120582 119906) is a solution of (34) Combining this and(44) and using the fact that the eigenfunctions correspondingto the eigenvalue 120582

119895of the operator 119870 with 119895 ge 2 have

Abstract and Applied Analysis 5

to change its sign in Ω it concludes that Case (ii) andCase (iii) cannot occurTherefore there exists an unboundedconnected subset C+ of the set

(120582 119906) isin (0infin) times 119875 119906 = 119860 (120582 119906) 119906 isin int119875 cup (12058201 0)

(48)

such that (12058201 0) isin C+

Proof of Theorem 1 It is clear that any solution of (34) of theform (120582 119906) yields a solution 119906 of (1) We will show that C+joins (1205820

1 0) to (120582infin

1infin)

Let (120583119899 119910119899) isin C+ satisfy

120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817 997888rarr infin (49)

We note that 120583119899gt 0 for all 119899 isin N since 119906 = 0 is the only

solution of (34) (ie (26)) for 120582 = 0In fact suppose on the contrary that 119906 is a nontrivial

solution of the problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(50)

then 119906 satisfies the linear problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ (119909) 119906 = 0 on 120597Ω

(51)

where (119909) = 119887(119909)(119892(119906)119906) Together with (H2) and theresults of Krasnoselrsquoskii [1] it follows that 119906 equiv 0 in Ω whichis a contradiction Therefore (34) (ie (26)) with 120582 = 0 hasonly trivial solution

Case 1 (120582infin1lt 120582 lt 120582

0

1) In this case we show that

(120582infin

1 1205820

1) sube (120582 isin R | exist (120582 119906) isin C

+) (52)

We divide the proof into two steps

Step 1We show that if there exists a constant number119872 gt 0

such that

120583119899sub (0119872] (53)

then C+ joins (12058201 0) to (120582infin

1infin)

From (53) we have 119910119899 rarr infin We divide the equation

119910119899= 120583119899Kinfin[119886 (119891infin119910119899+ ℎ (119910

119899))] +R

infin[119887120591 (minus119896 (119910

119899))]

in 119862 (Ω) (54)

by 119910119899 and set V

119899= 119910119899119910119899 Since V

119899is bounded in 119883

choosing a subsequence and relabeling if necessary we seethat V

119899rarr V for some V isin 119883 with V = 1 Moreover

from (31) and the fact that ℎ and are nondecreasing togetherwith the assertion 119910

119899 rarr infin there exists some 119903

119899119895le 119910119899

such that ℎ(119903119899119895) ge ℎ119910

119899 and 119903

119899119895rarr infin It follows that

ℎ(119910119899)119910119899 le ℎ(119903

119899119895)119903119899119895

rarr 0 as 119899 rarr infin Subsequentlywe have

lim119899rarrinfin

1003816100381610038161003816ℎ (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= lim119899rarrinfin

1003816100381610038161003816119896 (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= 0 (55)

since |ℎ(119910119899(119905))|119910

119899 le ℎ(|119910

119899(119905)|)119910

119899 le ℎ(119910

119899)119910119899 and

|119896(119910119899(119905))|119910

119899 le (|119910

119899(119905)|)119910

119899 le (119910

119899)119910119899 Therefore

V = 120583Kinfin[119886119891infinV] in 119862 (Ω) (56)

where 120583 = lim119899rarrinfin

120583119899 again choosing a subsequence and

relabeling if necessaryThus

minusΔV = 120583119886 (119909) 119891infinV in Ω

120597V120597n

+ 119887 (119909) 119892infinV = 0 on 120597Ω(57)

Since V = 1 and V ge 0 the strong positivity ofKinfin

ensuresthat

V gt 0 on Ω (58)

Thus 120583 = 120582infin1 and accordinglyC joins (1205820

1 0) to (120582infin

1infin)

Step 2We show that there exists a constant119872 such that 120583119899isin

(0119872] for all 119899From (H1) and (H2) there exist constants 120572 120573 isin (0infin)

such that

119891 (119906) ge 120572119906 119906 isin [0infin)

119892 (119906) le 120573119906 119906 isin [0infin)

(59)

By the same method to defineKinfinandR

infinin Section 3

we may defineKlowast andRlowast as followsLetKlowast 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear

boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(60)

Then Klowast is uniquely extended to a linear mapping of 119862(Ω)compactly into 1198621(Ω) and it is strongly positive Let Rlowast 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear boundary

value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 120595 on 120597Ω

(61)

Let 120582lowast1be the eigenvalue of the linear problem

minusΔ119906 = 120582119886 (119909) 120572119906 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(62)

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Then ℎ and are nondecreasing and there exist 119903119899 infin such

that

lim119899rarrinfin

ℎ (119903119899)

119903119899

= lim119899rarrinfin

(119903119899)

119903119899

= 0 (33)

Indeed for any 119903 gt 0 there exists 119904 le 119903 such that ℎ(119903)119903 =|ℎ(119904)|119903 Additionally if we assume 119904 le 119862 for some 119862 gt 0then it follows that ℎ(119903)119903 le 119862119903 rarr 0 as 119903 rarr infin On theother hand we assume that there exists 119903

119899such that 119904

119899rarr infin

then it follows from (31) that ℎ(119903119899)119903119899le |ℎ(119904

119899)|119904119899rarr 0 as

119895 rarr infin as desiredWe consider

119906 = 120582K0[119886 (1198910119906 + 120585 (119906))] +R

0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(34)

as a bifurcation problem from the trivial solution 119906 equiv 0Define119870 119883 rarr 119883

119870119906 (119905) = K0[1198861198910119906] (119905) (35)

then 119870 is a strongly positive linear operator on 119883 It is easyto verify that119870 119875 rarr 119875 is completely continuous From [14Theorem 32] it follows that

119903 (119870) = [1205820

1]minus1

(36)

DefineN [0infin) times 119883 rarr 119883 by

N (120582 119906) = 120582K0 [119886120585 (119906)] +R0[119887120591 (minus120577 (119906))] in 119862 (Ω)

(37)

then we have from (30) that

N (120582 119906) = ∘ (119906) (38)

locally and uniformly in 120582It is very easy to check that (34) enjoys the structural

requirements for applying the unilateral global bifurcationtheory of [12 Sections 64 65] (by a counter example ofDancer [16] the global unilateral theorem of Rabinowitz [8]is false as stated So it cannot be used) As the theoremof Crandall and Rabinowitz [17] is applied to get the localbifurcation to positive solutions from (120582

0

1 0) the algebraic

multiplicity of Esquinas and Lopez-Gomez [18] (see [19Chapter 4]) equals 1 and therefore by [12 Theorem 562]or [19 Proposition 1231] the local index of 0 as a fixedpoint of 119868 minus 120582K

0changes sign as 120582 crosses 1205820

1 Therefore it

follows from Lemma 7 that there exists the component C+that satisfies one of the following

(i) C+ is unbounded in R times 119880

(ii) there exists 120582lowast isin (0infin) with 120582lowast = 1205820

1and 120595 isin 119862(Ω)

which changes its sign onΩ such that

[119868 minus 120582lowastK0] 120595 = 0 (120582

lowast 0) isin C

+ (39)

(iii) C+ contains a point

(120582 119910) isin 119877 times (119884 0) (40)

where 119884 is the complement of ker[119868119883minus 1205820

1K0(119886(sdot))]

in119883

In what follows we will show that the above Case (ii) andCase (iii) do not occur

In fact if (120583 119910) isin C+ is a nontrivial solution of (34) then119910 satisfies the problem

minusΔ119910 = 120583119886 (119909) 119891 (119910) in Ω

120597119910

120597n+ 119887 (119909) 119892 (119910) = 0 on 120597Ω

(41)

that is to say 119910 satisfies the linear problem

minusΔ119910 = 120583119886 (119909) 119891 (119909) 119910 in Ω

120597119910

120597n+ 119887 (119909) 119892 (119909) 119910 = 0 on 120597Ω

(42)

where

119892 (119909) =

119892 (119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198920 as 119910 (119909) = 0

119891 (119909) =

119891(119910 (119909))

119910 (119909) as 119910 (119909) = 0

1198910 as 119910 (119909) = 0

(43)

We claim that

119910 isin C+ 119910 = 0 997904rArr 119910 isin int119875 cup int (minus119875) (44)

Suppose on the contrary that (120583 119910) isin C+ with 119910 = 0 and119910 isin 120597119875Then there exists a sequence (120583

119899 119906119899) sub C+cap119875with

119906119899gt 0 for 119899 isin N such that

(120583119899 119906119899) 997888rarr (120583 119910) as 119899 997888rarr infin (45)

and consequently

119910 (119909) ge 0 119909 isin Ω (46)

Combining this with the fact that (120583 119910) is a nontrivialsolution of (34) and using the strongmaximumprinciple [20Theorem 24] and (42) it concludes that

119910 (119909) gt 0 119909 isin Ω (47)

contracting 119910 isin 120597119875Similarly for (120578 119910) isin C+ with 119910 = 0 and 119910 isin 120597(minus119875) we

get the desired contradictionTherefore the claim (44) is trueSince 119891 and 119892 are odd in R (120582 minus119906) is a solution of (34)

if and only if (120582 119906) is a solution of (34) Combining this and(44) and using the fact that the eigenfunctions correspondingto the eigenvalue 120582

119895of the operator 119870 with 119895 ge 2 have

Abstract and Applied Analysis 5

to change its sign in Ω it concludes that Case (ii) andCase (iii) cannot occurTherefore there exists an unboundedconnected subset C+ of the set

(120582 119906) isin (0infin) times 119875 119906 = 119860 (120582 119906) 119906 isin int119875 cup (12058201 0)

(48)

such that (12058201 0) isin C+

Proof of Theorem 1 It is clear that any solution of (34) of theform (120582 119906) yields a solution 119906 of (1) We will show that C+joins (1205820

1 0) to (120582infin

1infin)

Let (120583119899 119910119899) isin C+ satisfy

120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817 997888rarr infin (49)

We note that 120583119899gt 0 for all 119899 isin N since 119906 = 0 is the only

solution of (34) (ie (26)) for 120582 = 0In fact suppose on the contrary that 119906 is a nontrivial

solution of the problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(50)

then 119906 satisfies the linear problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ (119909) 119906 = 0 on 120597Ω

(51)

where (119909) = 119887(119909)(119892(119906)119906) Together with (H2) and theresults of Krasnoselrsquoskii [1] it follows that 119906 equiv 0 in Ω whichis a contradiction Therefore (34) (ie (26)) with 120582 = 0 hasonly trivial solution

Case 1 (120582infin1lt 120582 lt 120582

0

1) In this case we show that

(120582infin

1 1205820

1) sube (120582 isin R | exist (120582 119906) isin C

+) (52)

We divide the proof into two steps

Step 1We show that if there exists a constant number119872 gt 0

such that

120583119899sub (0119872] (53)

then C+ joins (12058201 0) to (120582infin

1infin)

From (53) we have 119910119899 rarr infin We divide the equation

119910119899= 120583119899Kinfin[119886 (119891infin119910119899+ ℎ (119910

119899))] +R

infin[119887120591 (minus119896 (119910

119899))]

in 119862 (Ω) (54)

by 119910119899 and set V

119899= 119910119899119910119899 Since V

119899is bounded in 119883

choosing a subsequence and relabeling if necessary we seethat V

119899rarr V for some V isin 119883 with V = 1 Moreover

from (31) and the fact that ℎ and are nondecreasing togetherwith the assertion 119910

119899 rarr infin there exists some 119903

119899119895le 119910119899

such that ℎ(119903119899119895) ge ℎ119910

119899 and 119903

119899119895rarr infin It follows that

ℎ(119910119899)119910119899 le ℎ(119903

119899119895)119903119899119895

rarr 0 as 119899 rarr infin Subsequentlywe have

lim119899rarrinfin

1003816100381610038161003816ℎ (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= lim119899rarrinfin

1003816100381610038161003816119896 (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= 0 (55)

since |ℎ(119910119899(119905))|119910

119899 le ℎ(|119910

119899(119905)|)119910

119899 le ℎ(119910

119899)119910119899 and

|119896(119910119899(119905))|119910

119899 le (|119910

119899(119905)|)119910

119899 le (119910

119899)119910119899 Therefore

V = 120583Kinfin[119886119891infinV] in 119862 (Ω) (56)

where 120583 = lim119899rarrinfin

120583119899 again choosing a subsequence and

relabeling if necessaryThus

minusΔV = 120583119886 (119909) 119891infinV in Ω

120597V120597n

+ 119887 (119909) 119892infinV = 0 on 120597Ω(57)

Since V = 1 and V ge 0 the strong positivity ofKinfin

ensuresthat

V gt 0 on Ω (58)

Thus 120583 = 120582infin1 and accordinglyC joins (1205820

1 0) to (120582infin

1infin)

Step 2We show that there exists a constant119872 such that 120583119899isin

(0119872] for all 119899From (H1) and (H2) there exist constants 120572 120573 isin (0infin)

such that

119891 (119906) ge 120572119906 119906 isin [0infin)

119892 (119906) le 120573119906 119906 isin [0infin)

(59)

By the same method to defineKinfinandR

infinin Section 3

we may defineKlowast andRlowast as followsLetKlowast 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear

boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(60)

Then Klowast is uniquely extended to a linear mapping of 119862(Ω)compactly into 1198621(Ω) and it is strongly positive Let Rlowast 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear boundary

value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 120595 on 120597Ω

(61)

Let 120582lowast1be the eigenvalue of the linear problem

minusΔ119906 = 120582119886 (119909) 120572119906 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(62)

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

to change its sign in Ω it concludes that Case (ii) andCase (iii) cannot occurTherefore there exists an unboundedconnected subset C+ of the set

(120582 119906) isin (0infin) times 119875 119906 = 119860 (120582 119906) 119906 isin int119875 cup (12058201 0)

(48)

such that (12058201 0) isin C+

Proof of Theorem 1 It is clear that any solution of (34) of theform (120582 119906) yields a solution 119906 of (1) We will show that C+joins (1205820

1 0) to (120582infin

1infin)

Let (120583119899 119910119899) isin C+ satisfy

120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817 997888rarr infin (49)

We note that 120583119899gt 0 for all 119899 isin N since 119906 = 0 is the only

solution of (34) (ie (26)) for 120582 = 0In fact suppose on the contrary that 119906 is a nontrivial

solution of the problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 119892 (119906) = 0 on 120597Ω

(50)

then 119906 satisfies the linear problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ (119909) 119906 = 0 on 120597Ω

(51)

where (119909) = 119887(119909)(119892(119906)119906) Together with (H2) and theresults of Krasnoselrsquoskii [1] it follows that 119906 equiv 0 in Ω whichis a contradiction Therefore (34) (ie (26)) with 120582 = 0 hasonly trivial solution

Case 1 (120582infin1lt 120582 lt 120582

0

1) In this case we show that

(120582infin

1 1205820

1) sube (120582 isin R | exist (120582 119906) isin C

+) (52)

We divide the proof into two steps

Step 1We show that if there exists a constant number119872 gt 0

such that

120583119899sub (0119872] (53)

then C+ joins (12058201 0) to (120582infin

1infin)

From (53) we have 119910119899 rarr infin We divide the equation

119910119899= 120583119899Kinfin[119886 (119891infin119910119899+ ℎ (119910

119899))] +R

infin[119887120591 (minus119896 (119910

119899))]

in 119862 (Ω) (54)

by 119910119899 and set V

119899= 119910119899119910119899 Since V

119899is bounded in 119883

choosing a subsequence and relabeling if necessary we seethat V

119899rarr V for some V isin 119883 with V = 1 Moreover

from (31) and the fact that ℎ and are nondecreasing togetherwith the assertion 119910

119899 rarr infin there exists some 119903

119899119895le 119910119899

such that ℎ(119903119899119895) ge ℎ119910

119899 and 119903

119899119895rarr infin It follows that

ℎ(119910119899)119910119899 le ℎ(119903

119899119895)119903119899119895

rarr 0 as 119899 rarr infin Subsequentlywe have

lim119899rarrinfin

1003816100381610038161003816ℎ (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= lim119899rarrinfin

1003816100381610038161003816119896 (119910119899 (119905))1003816100381610038161003816

10038171003817100381710038171199101198991003817100381710038171003817

= 0 (55)

since |ℎ(119910119899(119905))|119910

119899 le ℎ(|119910

119899(119905)|)119910

119899 le ℎ(119910

119899)119910119899 and

|119896(119910119899(119905))|119910

119899 le (|119910

119899(119905)|)119910

119899 le (119910

119899)119910119899 Therefore

V = 120583Kinfin[119886119891infinV] in 119862 (Ω) (56)

where 120583 = lim119899rarrinfin

120583119899 again choosing a subsequence and

relabeling if necessaryThus

minusΔV = 120583119886 (119909) 119891infinV in Ω

120597V120597n

+ 119887 (119909) 119892infinV = 0 on 120597Ω(57)

Since V = 1 and V ge 0 the strong positivity ofKinfin

ensuresthat

V gt 0 on Ω (58)

Thus 120583 = 120582infin1 and accordinglyC joins (1205820

1 0) to (120582infin

1infin)

Step 2We show that there exists a constant119872 such that 120583119899isin

(0119872] for all 119899From (H1) and (H2) there exist constants 120572 120573 isin (0infin)

such that

119891 (119906) ge 120572119906 119906 isin [0infin)

119892 (119906) le 120573119906 119906 isin [0infin)

(59)

By the same method to defineKinfinandR

infinin Section 3

we may defineKlowast andRlowast as followsLetKlowast 119862120579(Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear

boundary value problem

minusΔ119906 = 120601 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(60)

Then Klowast is uniquely extended to a linear mapping of 119862(Ω)compactly into 1198621(Ω) and it is strongly positive Let Rlowast 1198621+120579(120597Ω) rarr 119862

2+120579(Ω) be the resolvent of the linear boundary

value problem

minusΔ119906 = 0 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 120595 on 120597Ω

(61)

Let 120582lowast1be the eigenvalue of the linear problem

minusΔ119906 = 120582119886 (119909) 120572119906 in Ω

120597119906

120597n+ 119887 (119909) 120573119906 = 0 on 120597Ω

(62)

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

and let 120595lowast isin 119875 be the corresponding eigenfunction Then

minusΔ120595lowast= 120582lowast

1119886 (119909) 120572120595

lowast in Ω

120597120595lowast

120597n+ 119887 (119909) 120573120595

lowast= 0 on 120597Ω

(63)

Since (120583119899 119910119899) isin C+ satisfies (1) and (120582lowast

1 120595lowast) satisfies (63) it

follows from (59) and Greenrsquos formula that

(120583119899minus 120582lowast

1) intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

le 120583119899intΩ

119886 (119909) 119891 (119906119899 (119909)) 120595lowast(119909) 119889119909

minus 120582lowast

1intΩ

119886 (119909) 120572120595lowast(119909) 119906119899 (119909) 119889119909

= intΩ

Δ120595lowast(119909) 119906119899 (119909) 119889119909 minus int

Ω

Δ119906119899 (119909) 120595

lowast(119909) 119889119909

= int120597Ω

119906119899

120597120595lowast

120597n119889120590 minus int

120597Ω

120595lowast 120597119906119899

120597n119889120590

= minusint120597Ω

119887 (119909) 120573120595lowast119906119899119889120590 + int

120597Ω

119887 (119909) 119892 (119906119899) 120595lowast119889120590

le int120597Ω

119887 (119909) [120573119906119899120595lowastminus 120573120595lowast119906119899] 119889120590 = 0

(64)

and here 119889120590 is the surface element of 120597Ω Subsequently 120583119899le

120582lowast

1 Therefore the component C+ joins (1205820

1 0) to (120582infin

1infin)

Case 2 (12058201lt 120582 lt 120582

infin

1) In this case if (120583

119899 119910119899) isin C+ is such

that

lim119899rarrinfin

(120583119899+1003817100381710038171003817119910119899

1003817100381710038171003817) = infin

lim119899rarrinfin

120583119899= infin

(65)

then

(1205820

1 120582infin

1) sube 120582 isin (0infin) | (120582 119906) isin C

+ (66)

and moreover

(120582 times 119883) cap C+= 0 (67)

Assume that there exists119872 gt 0 such that for all 119899 isin N

120583119899isin (0119872] (68)

Applying a similar argument to that used in Step 1 of Case1 after taking a subsequence and relabeling if necessary itfollows that

(120583119899 119910119899) 997888rarr (120582

infin

1infin) 119899 997888rarr infin (69)

AgainC+ joins (12058201 0) to (120582infin

1infin) and the result follows

Proof of Corollary 2 It is a directly desired consequence ofTheorem 1

Conflict of Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the anonymous refereesfor their valuable suggestions This work was supportedby the NSFC (no 11361054 and no 11201378) SRFDP (no20126203110004) and Gansu provincial National ScienceFoundation of China (no 1208RJZA258)

References

[1] M A Krasnoselrsquoskii Positive Solutions of Operator Equations PNoordhoff Groningen The Netherlands 1964

[2] A Ambrosetti D Arcoya and B Buffoni ldquoPositive solutions forsome semi-positone problems via bifurcation theoryrdquo Differen-tial and Integral Equations vol 7 no 3-4 pp 655ndash663 1994

[3] K Umezu ldquoGlobal positive solution branches of positoneproblemswith nonlinear boundary conditionsrdquoDifferential andIntegral Equations vol 13 no 4ndash6 pp 669ndash686 2000

[4] K Umezu ldquoBifurcation from infinity for asymptotically linearelliptic eigenvalue problemsrdquo Journal of Mathematical Analysisand Applications vol 267 no 2 pp 651ndash664 2002

[5] J Henderson and H Wang ldquoPositive solutions for nonlineareigenvalue problemsrdquo Journal of Mathematical Analysis andApplications vol 208 no 1 pp 252ndash259 1997

[6] RMa ldquoExistence of positive solutions of a fourth-order bound-ary value problemrdquoAppliedMathematics and Computation vol168 no 2 pp 1219ndash1231 2005

[7] R Ma and B Thompson ldquoNodal solutions for nonlineareigenvalue problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 5 pp 707ndash718 2004

[8] P H Rabinowitz ldquoSome global results for nonlinear eigenvalueproblemsrdquo Journal of Functional Analysis vol 7 pp 487ndash5131971

[9] J Shi ldquoExact multiplicity of solutions to superlinear andsublinear problemsrdquo Nonlinear Analysis Theory Methods ampApplications vol 50 no 5 pp 665ndash687 2002

[10] A Ambrosetti and P Hess ldquoPositive solutions of asymptoticallylinear elliptic eigenvalue problemsrdquo Journal of MathematicalAnalysis and Applications vol 73 no 2 pp 411ndash422 1980

[11] R Precup ldquoMoser-Harnack inequality Krasnoselrsquoskii type fixedpoint theorems in cones and elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 40 no 2 pp 301ndash313 2012

[12] J Lopez-Gomez Spectral Theory and Nonlinear FunctionalAnalysis vol 426 of CRC Research Notes in MathematicsChapman amp Hall Boca Raton Fla USA 2001

[13] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Springer New York NY USA 1983

[14] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[15] HAmann ldquoNonlinear elliptic equationswith nonlinear bound-ary conditionsrdquo in New Developments in Differential EquationsW Eckhaus Ed vol 21 of Mathematical Studies pp 43ndash63North-Holland Amsterdam The Netherlands 1976

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

[16] E N Dancer ldquoBifurcation from simple eigenvalues and eigen-values of geometric multiplicity onerdquoTheBulletin of the LondonMathematical Society vol 34 no 5 pp 533ndash538 2002

[17] M G Crandall and P H Rabinowitz ldquoBifurcation from simpleeigenvaluesrdquo Journal of Functional Analysis vol 8 pp 321ndash3401971

[18] J Esquinas and J Lopez-Gomez ldquoOptimal multiplicity in localbifurcation theory I Generalized generic eigenvaluesrdquo Journalof Differential Equations vol 71 no 1 pp 72ndash92 1988

[19] J Lopez-Gomez and C Mora-Corral Algebraic Multiplicity ofEigenvalues of Linear Operators vol 177 of Operator TheoryAdvances andApplications Birkhauser Basel Switzerland 2007

[20] Y H DuOrder Structure and Topological Methods in NonlinearPDEs vol 1 Maximum Principles and Applications of Series onPartial Differential Equations andApplicationsWorld ScientificSingapore 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of