Nonlinear Analysis 109 (2014) 148–155

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Singular quasilinear elliptic problems on unbounded domainsPavel Drábek ∗, Lakshmi SankarDepartment of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzeň,Czech Republic

a r t i c l e i n f o

Article history:Received 7 April 2014Accepted 2 July 2014Communicated by Enzo Mitidieri

MSC:35J2535J7535J92

Keywords:p-LaplacianUnbounded domainsInfinite semipositoneSub and supersolutionsSingular problems

a b s t r a c t

Weprove the existence of a solution between an ordered pair of sub and supersolutions forsingular quasilinear elliptic problems on unbounded domains. Further, we use this resultto establish the existence of a positive solution to the problem

−∆pu = λK(x)f (u) in Bc1,

u = 0 on ∂B1,

u(x)→0 as |x| → ∞,

where Bc1 = x ∈ Rn

| |x| > 1, ∆pu = div(|∇u|p−2∇u), 1 < p < n, λ is a positive

parameter, K belongs to a class of functions which satisfy certain decay assumptions andf belongs to a class of (p − 1)-subhomogeneous functions which may be singular at theorigin, namely lims→0+ f (s) = −∞. Our methods can be also applied to establish a similarexistence result when the domain is entire Rn.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

We consider problems of the form−∆pu = h(x, u) inΩc,u = 0 on ∂Ω, (1)

where ∆pu = div(|∇u|p−2∇u), p > 1,Ω ⊂ Rn is a simply connected bounded domain containing the origin with C2

boundary ∂Ω,Ωc= Rn

\Ω is an exterior domain, and h : Ωc× (0,∞) → R is a Caratheodory function, i.e, h(x, ·) is

continuous for a.e. x ∈ Ωc and h(·, s) is measurable for all s ∈ (0,∞). We prove the existence of a positive weak solution ofproblem (1) under the assumption of the existence of an ordered pair of sub and supersolutions. We will allow h(x, s) to besingular when x ∈ ∂Ω or when s = 0. A similar result was established in [1] in case of bounded domains. Our definitions ofa weak solution, sub and supersolutions are given below.

By a weak solution of (1), we mean a function u ∈ C(Ωc) ∩ C1(Ωc)which satisfiesΩc

|∇u|p−2∇u · ∇w =

Ωc

h(x, u)w for allw ∈ C∞

c (Ωc),

u = 0 on ∂Ω.(2)

Here, C∞c (Ω

c) denotes the set of all smooth functions with compact support inΩc .

∗ Corresponding author. Tel.: +420 377632648.E-mail addresses: pdrabek@kma.zcu.cz (P. Drábek), lakshmi@ntis.zcu.cz (L. Sankar).

http://dx.doi.org/10.1016/j.na.2014.07.0020362-546X/© 2014 Elsevier Ltd. All rights reserved.

P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155 149

By a subsolution of (1), we mean a function ψ ∈ C(Ωc) ∩ C1(Ωc) that satisfiesΩc

|∇ψ |p−2

∇ψ · ∇w ≤

Ωc

h(x, ψ)w for everyw ∈ C∞

c (Ωc), w ≥ 0 inΩc,

ψ > 0 inΩc,ψ = 0 on ∂Ω.

By a supersolution of (1), we mean a function Z ∈ C(Ωc) ∩ C1(Ωc) that satisfiesΩc

|∇Z |p−2

∇Z · ∇w ≥

Ωc

h(x, Z)w for everyw ∈ C∞

c (Ωc), w ≥ 0 inΩc,

Z > 0 inΩc,Z = 0 on ∂Ω.

Then we prove the following result.

Theorem 1.1. Let ψ be a subsolution of (1) and Z be a supersolution of (1) such that ψ ≤ Z inΩc and let h satisfy:

(H0) for each bounded set M such that M ⊂ Ωc , there exists a constant KM > 0 such that for a. e. x ∈ M and for all s satisfyingψ(x) ≤ s ≤ Z(x), |h(x, s)| ≤ KM .

Then (1) has a weak solution u ∈ C1,βloc (Ω

c), for some β ∈ (0, 1) such that ψ ≤ u ≤ Z inΩc .

We note here that the conditions ψ, Z > 0 inΩc, ψ = Z = 0 on ∂Ω are due to the singularity of h, and can be relaxedto the usual conditions ψ ≤ 0, Z ≥ 0 on ∂Ω in the non singular case. The details of this can be found in the proof ofTheorem 1.1. Our methods can be also extended to study the existence of positive weak solutions to problems of the form

−∆pu = h(x, u) in Rn, (3)

where h satisfies (H0) on Rn. Here, by a weak solution of (3), wemean a function u ∈ C1(Rn)which satisfies a correspondingextension of (2) to Rn. The definitions of sub and supersolutions remain the same except that we do not have to deal with aboundary condition. We obtain:

Theorem 1.2. Let ψ be a subsolution of (3) and Z be a supersolution of (3) such that ψ ≤ Z in Rn and let h satisfy (H0) on Rn.Then (3) has a weak solution u ∈ C1,β

loc (Rn), for some β ∈ (0, 1) such that ψ ≤ u ≤ Z in Rn.

As an application, we consider problems of the form−∆pu = λK(x)f (u) in Bc

1,u = 0 on ∂B1,u(x) → 0 as |x| → ∞,

(4)

where Bc1 = x ∈ Rn

| |x| > 1, 1 < p < n, λ is a positive parameter, f : (0,∞) → R belongs to a class of continuousfunctions which are (p− 1)-subhomogeneous andmay be singular at the origin, and K belongs to a class of functions whichsatisfy certain decay assumptions. Similar problems were studied recently in [2,3], where the authors used Kelvin transfor-mation to reduce (4) to a two point boundary value problem and established the existence of positive radial solutions. Inour paper, we study (4) in its original setting in a wider class of possibly non radial functions and establish the existence ofa positive weak solution.

Study of positive solutions to boundary value problems of the form−∆pu = λf (u) inΩ,u = 0 on ∂Ω,

whereΩ is a smooth domain in Rn, λ is a positive parameter and f ∈ C((0,∞),R) has been of great interest over the years.The case when f (0) < 0 is referred in the literature as semipositone and it is known to be mathematically challenging toestablish the existence of positive solutions.When the domain is bounded, see [4–10] for some existence results in this case.Existence results have been also established in the case when f is singular at the origin, i.e., lims→0+ f (s) = −∞ (known asan infinite semipositone problem, see [11–13,1,14–17]).When the domain is unbounded there are very few results availablein this direction.

We now state our assumptions on f and K .

(H1) lims→∞ f (s) = ∞.(H2) lims→∞

f (s)sp−1 = 0 ((p − 1)-subhomogeneity of f ).

(H3) There exist b > 0, β ∈ (0, 1) such that f (s) ≥−bsβ ∀ s ≥ 0.

(H4) There exist ϵ > 0, A > 0 such that f (s) ≤ A ∀s ∈ (0, ϵ).(H5) There exist C0 > 0, α > n + β(

n−pp−1 ) such that 0 < K(x) < C0

|x|α ∀x ∈ Bc1.

(H6) inf|x|=r K(x) > 0 ∀r ≥ 1.

150 P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155

We include the possibility of f being singular at the origin, namely, lims→0+ f (s) = −∞, but the singularity is controlledby assumption (H3). The decay assumption on K , (H5), is as in [2,3] and this implies that K ∈ L∞(Bc

1)∩ Lγ (Bc1), for all γ >

nα.

When f has no singularity, i.e, when β = 0, (H5) reduces to 0 < K(x) < C0|x|α in Bc

1 for some α > n. In particular, (H5) doesnot depend on p > 1 in this case.

We have the following existence result on Bc1.

Theorem 1.3. Let f satisfy (H1)–(H4) and K satisfy (H5)–(H6). Then there existsλ0 > 0 such that for anyλ ≥ λ0 problem (4) hasa positive weak solution u ∈ C1,β

loc (Bc1), for some β ∈ (0, 1).

We also establish the existence of a positive weak solution to the problem on entire Rn,−∆pu = λK(x)f (u) in Rn,u(x) → 0 as |x| → ∞,

(5)

where λ, f are as before and K ∈ L∞(Rn) satisfies the hypotheses below:

(H5) There exist C0 > 0, α > n + β(n−pp−1 ) such that 0 < K(x) < C0

|x|α ∀x ∈ Rn.

(H6) inf|x|=r K(x) > 0 ∀r ≥ 0.

We obtain:

Theorem 1.4. Let f satisfy (H1)–(H4) and K ∈ L∞(Rn) satisfy (H5), (H6). Then there exists λ0 > 0 such that for any λ ≥ λ0problem (5) has a positive weak solution u ∈ C1,β

loc (Rn), for some β ∈ (0, 1).

In Section 2 we provide the proofs of Theorems 1.1 and 1.2. Some a priori estimates which are useful in the constructionof sub and supersolutions associated with (4) and (5) are provided in Section 3. In Sections 4 and 5 we provide the proofsof Theorems 1.3 and 1.4, respectively. It is important to note here that in case of infinite semipositone problems, thedifficulty lies in the construction of a suitable subsolution. The reason lies in the fact that the subsolutionψ needs to satisfylimx→∂B1 −∆pψ = −∞ on one hand, and −∆pψ > 0 in a large part of the interior of Bc

1 on the other hand.

2. Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1. Choose a sequence of subdomains Ωj∞

j=1 ofΩc such thatΩ1 ⊂ Ω2 ⊂ Ω2 ⊂ · · · ⊂ Ω j ⊂ Ωj+1 ⊂

Ω j+1 · · · ⊂ Ωc and ∪∞

j=1Ωj = Ωc . For fixed j ∈ N, consider the problem−∆pu = h(x, u) inΩj,u(x) = ψ(x) on ∂Ωj.

(6)

Let aj = minΩ jψ > 0 and bj = maxΩ j

Z and for x ∈ Ωj, define

hj(x, u) =

h(x, aj), u < ajh(x, u), aj ≤ u ≤ bjh(x, bj), u > bj.

Restrictions ofψ and Z onΩj (denoted asψj, Zj, respectively) are clearly the sub and supersolutions (in the sense of definitionin [18]) of

−∆pu = hj(x, u) inΩj,u(x) = ψ(x) on ∂Ωj.

(7)

Since h(x, s) satisfies the assumption (H0), the hypotheses of Theorem 3.1 in [18] are fulfilled with Ω = Ωj and f = hj.Hence (7) has a minimal solution uj ∈ C1,α(Ω j) for some α = α(j) such that ψj(x) ≤ uj(x) ≤ Zj(x) for all x ∈ Ωj. Clearly ujis also a solution of (6).

We extend uj to entireΩc by setting uj(x) = ψ(x) for all x ∈ Ωc\Ωj. Then uj ∈ C(Ωc) and satisfies

ψ(x) ≤ uj(x) ≤ Z(x), for all x ∈ Ωc and j ≥ 1. (8)

Next we prove that, for every fixed k ≥ 1, there exist constants ck > 0 and β = β(k) ∈ (0, 1) such that for all j ≥ k + 1,we have

∥uj∥C1,β (Ωk)≤ ck.

Indeed, (8) and (H0) imply that there exists dk > 0 such that

∥h(·, uj(·))∥L∞(Ωk+1) ≤ dk, for all j ≥ k + 1.

P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155 151

Our claim now follows from the application of Proposition 3.7 in [19] to the equation

−∆puj = h(x, uj(x)), x ∈ Ωk+1.

Due to the compact embedding C1,β(Ωk) →→ C1(Ωk) for each fixed k ≥ 1, uj contains a subsequence, renamed asuk

j such that ukj |Ωk

converges to uk∈ C1(Ωk) in C1 topology. Above arguments apply also to the sequence uk

j (insteadof uj) andΩk+1 (instead ofΩk). Hence uk

j contains a subsequence, renamed as uk+1j such that uk+1

j |Ωk+1converges to

uk+1∈ C1(Ωk+1) in C1 topology. Clearly uk+1(x) = uk(x) for x ∈ Ωk. Letting k → ∞, we can define a function u ∈ C1(Ωc)

as u(x) = uk(x) if x ∈ Ωk for all k ≥ 1.For each k, sinceΩk ⊂ Ωk+1,

Ωk

|∇ukj |

p−2∇uk

j · ∇w =

Ωk

h(x, ukj )w,

for everyw ∈ C∞c (Ωk) and for every j ≥ k + 1. By taking the limit of the sequence uk

j as j → ∞,Ωk

|∇uk|p−2

∇uk· ∇w =

Ωk

h(x, uk)w, (9)

for everyw ∈ C∞c (Ωk).

For each fixedw ∈ C∞c (Ω

c), there exists kw ≥ 1 such that suppw ⊂ Ωk for all k ≥ kw . Since (9) holds for every k ≥ kw ,and uk(x) = u(x) on support ofw,

Ωc|∇u|p−2

∇u · ∇w =

Ωc

h(x, u)w.

Clearly u satisfies ψ(x) ≤ u(x) ≤ Z(x) for all x ∈ Ωc , since ψ(x) ≤ uj(x) ≤ Z(x) for all j ≥ 1. Also u(x) = 0 on ∂Ωas ψ(x) = Z(x) = 0 on ∂Ω . Since u is uniformly bounded, it follows from [20] that u ∈ C(Ωc) ∩ C1,β

loc (Ωc), for some

β ∈ (0, 1).

Proof of Theorem 1.2. It can be easily verified that the proof of Theorem 1.2 follows along the same lines as in the aboveproof.

3. A priori estimates

Here we provide some preliminary results and important a priori estimates which will play important roles in theconstruction of sub and supersolutions in the next two sections.

Let K(x) = K(|x|) = inf|y|=r K(y) if |x| = r (K(x) > 0 by (H6)), and consider the eigenvalue problem−∆pu = λK(x) |u|p−2 u in Bc1,

u = 0 on ∂B1,u(x) → 0 as |x| → ∞.

(10)

The existence of the principal eigenvalue λ1 and corresponding eigenfunction φ1 ∈ C(Bc1) ∩ C1,β

loc (Bc1), β ∈ (0, 1) is estab-

lished in [21]. Some important properties of λ1 and φ1 are proved in [22,21] as well. The following lemma is from [21] andwill be used in the construction of the subsolution ψ in Section 3.

Lemma 3.1. Let φ1 be the eigenfunction corresponding to the first eigenvalue of (10). Then the following estimates hold for φ1.

(i) φ1 > 0 in Bc1.

(ii) (Decay near infinity) There exist 0 < C1 < C2 such that for all x ∈ Bc2

C1

|x|n−pp−1

≤ φ1(x) ≤C2

|x|n−pp−1.

(iii) (Gradient estimate near infinity) φ1 is radially symmetric (since K is radially symmetric) and there exist C1 > 0, and r0 > 1such that for all x ∈ Bc

r0 ,

C1

|x|n−1p−1

≤ |∇φ1(x)|.

(iv) (Gradient estimate at the boundary) There exists C2 > 0 such that for all x ∈ ∂B1,

|∇φ1(x)| ≤ C2.

152 P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155

Next we state some lemmas which are needed in the construction of our supersolution Z . The following result is aconsequence of Theorem 1 in [23], where a more general problem is discussed.

Lemma 3.2. Let u ∈ L∞(Ω) be a weak solution of

−∆pu = h(x, u) inΩ, 1 < p < n,

whereΩ ⊆ Rn. Assume also that h(x, u) ∈ Lγ1(Ω) for some γ1 > np . Let Br(x) denote the ball centered at x with radius r > 0.

Then for any x ∈ Ω

∥u∥L∞(Br (x)) ≤ C(∥u∥Lp∗ (B2r (x))+ ∥h(·, u)∥Lγ1 (B2r (x))),

where C = C(p, n, γ1) and B2r(x) ⊂ Ω .

Next we state a result from [22, Theorem 3, Theorem C], where the authors established decay estimates for solutions ofcertain p-Laplacian inequalities.

Lemma 3.3. Let l > n and 0 ≤ u ∈ W 1,ploc (B

c1) satisfy weakly 0 ≤ −∆pu(x) ≤ C0|x|−l in Bc

1 for some constant C0 > 0 andlim|x|→∞ u(x) = 0. Then there exist C1 > 0, C2 > 0 such that for all x ∈ Bc

2

C1

|x|n−pp−1

≤ u(x) ≤C2

|x|n−pp−1.

Using Lemmas 3.2 and 3.3, we prove the following result.

Lemma 3.4. The boundary value problem−∆pe = K(x), in Bc

1,e = 0 on ∂B1

(11)

has a weak solution e ∈ C(Bc1) ∩ C1,β

loc (Bc1), there exist C1 > 0, C2 > 0 such that for all x ∈ Bc

2

C1

|x|n−pp−1

≤ e(x) ≤C2

|x|n−pp−1, (12)

and there exists c > 0 such that for all x ∈ ∂B1, |∇e(x)| ≥ c > 0.

Proof. We first prove (11) has a weak solution e ∈ W 1,p0 (Bc

1). For this, we will show that K ∈ (W 1,p0 (Bc

1))∗, the dual of

W 1,p0 (Bc

1). Then by the properties of the p-Laplacian operator, (11) has a solution e ∈ W 1,p0 (Bc

1) (see [24]). Choose a testfunction v ∈ W 1,p

0 (Bc1). By the Sobolev embedding,W 1,p

0 (Bc1) → Lp

∗

(Bc1), v ∈ Lp

∗

(Bc1), where p∗

=npn−p is the critical Sobolev

exponent. By Holder’s inequality,K , v =

Bc1

K(x)v(x) ≤ ∥K∥Lq(Bc1)

∥v∥Lp∗ (Bc1)≤ ∥K∥Lq(Bc1)

∥v∥W1,p0 (Bc1)

, (13)

where q =np

np−n+p > nα, and

·, ·

is the duality pairing between (W 1,p

0 (Bc1))

∗ and W 1,p0 (Bc

1). Since K ∈ Lγ (Bc1) for all

γ > nα, ∥K∥Lq(Bc1)

< ∞. Thus from (13), we get K ∈ (W 1,p0 (Bc

1))∗.

Next we prove the estimate (12). From Theorem 2.5, (2.12) in [25], there exists a constant C > 0 such that

∥e∥p−1L∞(Bc1)

≤ C∥K∥L∞(Bc1)+ ∥e∥p−1

Lp(Bc1),

and thus e ∈ L∞(Bc1). Since K ∈ Lγ (Bc

1) for all γ >np and e ∈ L∞(Bc

1), Lemma 3.2 can be applied to e, and we get

∥e∥L∞(B1(x)) ≤ C(∥e∥Lp∗ (B2(x))+ ∥K∥Lγ (B2(x))), (14)

for all x ∈ Bc3. Note that by the Sobolev embedding, we have e ∈ Lp

∗

(Bc1). This fact together with (14) implies e(x) → 0 as

|x| → ∞. Since K satisfies (H5) and e(x) → 0 as |x| → ∞, (12) follows from Lemma 3.3. By regularity results in [26,20],e also belongs to C(Bc

1) ∩ C1,βloc (B

c1) for some β ∈ (0, 1). Also by applying Hopf’s maximum principle on the boundary ∂B1,

there exists a constant c > 0 such that for all x ∈ ∂B1, |∇e(x)| ≥ c > 0.

P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155 153

4. Existence result on Bc1

Let φ1 be the eigenfunction corresponding to the first eigenvalue λ1 of (10). We define a function ψ := λδφγ

1 , whereδ ∈ ( 1

p−1+β ,1

p−1 ], γ ∈ (1, (p−1)(α−p)(n−p)(p−1+β) ), and λ > 0 will be specified later. Notice that the last interval is nonempty because

of the condition on α in (H5). Then we can perform the following formal calculations:

∇ψ = λδγφγ−11 ∇φ1.

∆pψ = div(|∇ψ |p−2

∇ψ)

= λδ(p−1)γ p−1div(φ(γ−1)(p−1)1 |∇φ1|

p−2∇φ1)

= λδ(p−1)γ p−1∇(φ

(γ−1)(p−1)1 ) · |∇φ1|

p−2∇φ1 + φ

(γ−1)(p−1)1 ∆pφ1

= λδ(p−1)γ p−1

(γ − 1)(p − 1)φ(γ−1)(p−1)−1

1 |∇φ1|p− λ1φ

(γ−1)(p−1)1 φ

p−11 K(x)

.

Hence, formally,

−∆pψ = λδ(p−1)γ p−1λ1φγ (p−1)1 K(x)− λδ(p−1)γ p−1(γ − 1)(p − 1)

|∇φ1|p

φp−γ (p−1)1

. (15)

We now prove a lemma using which we will be able to show that the function ψ defined above is a subsolution of (4) forlarge values of λ.

Lemma 4.1. Let Br := x ∈ Rn| |x| < r denote the open ball of radius r centered at the origin. Then there exist r1 > 1 and

r2 ≫ 1 such that the following two inequalities hold.(i) There exists µ > 0 such that I = (γ − 1)(p − 1) |∇φ1|

p

φp−γ (p−1+β)1 K(x)

− λ1φγ (p−1)+γ β1 ≥ µ in (Br1\B1) ∪ Bc

r2 .

(ii) There exists m > 0 such that φ1 ≥ m in Br2\Br1 .Proof. By Hopf’s Maximum principle (see [27]) there exists c > 0 such that |∇φ1| ≥ c > 0 near ∂B1. Also φ1 ≈ 0 near ∂B1.Hence there exist r1 > 1, µ1 > 0 such that I ≥ µ1 in (Br1\B1).

Next we show that there exist r2 ≫ 1, µ2 > 0 such that I ≥ µ2 in Bcr2 . For this, we estimate the least value of

|∇φ1|p

φp−γ (p−1+β)1 K(x)

in Bcr , for all r ≥ max2, r0, where r0 is as in Lemma 3.1. Using (H5), (ii) and (iii) of Lemma 3.1,

|∇φ1|p

φp−γ (p−1+β)1 K(x)

≥C1|x|

(p−γ (p−1+β))n−pp−1

|x|α

C2C0|x|p( n−1

p−1 )

= C |x|α+(p−γ (p−1+β))n−pp−1

−p( n−1

p−1 ) , where C =C1

C2C0.

By the choice of γ < (p−1)(α−p)(n−p)(p−1+β) , it can be seen that the power of |x| is positive. Hence |∇φ1|

p

φp−γ (p−1+β)1 K(x)

is large for large values

of |x|. This together with the fact that φ1(x) → 0 as |x| → ∞ ensures the existence of r2 > max2, r0, µ2 > 0 such thatI ≥ µ2 in Bc

r2 . Thus for µ = minµ1, µ2, I ≥ µ in (Br1\B1) ∪ Bcr2 .

By the continuity and positivity of φ1 in Bc1, clearly there existsm > 0 such that φ1 ≥ m in Br2\Br1 .

Proof of Theorem 1.3. We first prove there exists a λ0 > 0 such that ψ is a subsolution of (4) for λ ≥ λ0. To prove this,according to (15) we need to achieve

λδ(p−1)γ p−1λ1φγ (p−1)1 K(x)− λδ(p−1)γ p−1(γ − 1)(p − 1)

|∇φ1|p

φp−γ (p−1)1

≤ λK(x)f (λδφγ1 ) (16)

in Bc1. Since K(x) ≤ K(x), (16) follows if we prove

λδ(p−1)γ p−1λ1φγ (p−1)1 − λδ(p−1)γ p−1(γ − 1)(p − 1)

|∇φ1|p

φp−γ (p−1)1 K(x)

≤ λf (λδφγ1 ) (17)

in Bc1. We now split the domain Bc

1 into three different regions Br1\B1, Br2\Br1 and Bcr2 , where r1, r2 are as in Lemma 4.1. Next

we will prove separately that (17) holds in Br2\Br1 and (Br1\B1) ∪ Bcr2 for large values of λ.

We first prove that there exists λ1 > 0 such that (17) holds in Br2\Br1 for λ ≥ λ1. Indeed, since φ1 ≥ m in Br2\Br1 , by(H1), we can choose a λ1 > 0 such that f (λδφγ1 ) ≥ γ p−1λ1 for all λ ≥ λ1. This and the fact that δ ≤

1p−1 imply

λδ(p−1)γ p−1λ1φγ (p−1)1 ≤ λf (λδφγ1 )

for λ ≥ λ1. Thus (17) holds in Br2\Br1 for λ ≥ λ1.

154 P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155

Next we prove that there exists λ2 > 0 such that for all λ ≥ λ2, (17) holds in (Br1\B1) ∪ Bcr2 . Indeed,

λδ(p−1)γ p−1λ1φγ (p−1)1 − λδ(p−1)γ p−1(γ − 1)(p − 1)

|∇φ1|p

φp−γ (p−1)1 K(x)

=λδ(p−1)γ p−1

φγ β

1

λ1φ

γ (p−1)+γ β1 − (γ − 1)(p − 1)

|∇φ1|p

φp−γ (p−1+β)1 K(x)

≤λδ(p−1)γ p−1

φγ β

1

(−µ) (since I ≥ µ in (Br1\B1) ∪ Bcr2 by Lemma 4.1)

=−bλδ(p−1)γ p−1

φγ β

1

µ

b

.

By the choice of δ, δ(p − 1) > 1 − δβ , and hence there exists a λ2 > 0 such that λδ(p−1)γ p−1(µ

b ) ≥ λ1−δβ for all λ ≥ λ2.Then for all λ ≥ λ2

−bλδ(p−1)γ p−1

φγ β

1

µ

b

≤

−bλ1−δβ

φγ β

1

=−bλ

λδβφγ β

1

≤ λf (λδφγ1 ) (by H3).

Thus (17) holds in (Br1\B1) ∪ Bcr2 for λ ≥ λ2. Let λ0 = maxλ1, λ2. Then ψ is a subsolution of (4) for all λ ≥ λ0.

Next, we construct a supersolution of (4). Define f (s) = maxu∈(0,s] f (u). Then f is non decreasing and satisfies (H2) sincef satisfies (H2) and (H4). Let e ∈ C(Bc

1) ∩ C1,βloc (B

c1) be as in Lemma 3.4. We define our supersolution to be Z = M(λ)e. Since

f satisfies (H2), for any λ ≥ λ0, there exists M(λ) ≫ 1 such that

1

∥e∥p−1∞ λ

≥f (M(λ)∥e∥∞)

(M(λ)∥e∥∞)p−1.

Then, formally,

−∆pZ = M(λ)p−1K(x) ≥ λK(x)f (M(λ)∥e∥∞) ≥ λK(x)f (Z).

Thus Z is a supersolution of (4) and decays to 0 by Lemma 3.4.For any λ ≥ λ0, by choosingM(λ) ≫ 1, we can ensure thatψ(x) ≤ Z(x) for every x ∈ Bc

1. Indeed, using the regularity ofφ1 up to ∂B1, and the fact that γ > 1, |∇ψ(x)| = 0, if x ∈ ∂B1. Also as mentioned in Lemma 3.4, e satisfies |∇e(x)| ≥ c > 0for every x ∈ ∂B1. Thus ifM(λ) is large, it is possible to obtainψ(x) ≤ Z(x) for every x ∈ B2\B1. Due to the decay estimates,(ii) in Lemma 3.1, (12) in Lemma 3.4, and the fact that γ > 1, we can also obtain ψ(x) ≤ Z(x) for every x ∈ Bc

2 if M(λ) islarge.

By assumption (H5), and the continuity of f on (0,∞), h(x, s) = λK(x)f (s), satisfies the hypotheses of Theorem 1.1. Thusby Theorem 1.1, (4) has a positive weak solution u ∈ C1,β

loc (Bc1), β ∈ (0, 1), such that ψ ≤ u ≤ Z . Clearly u(x) decays to 0 as

|x| → ∞.

5. Existence result on Rn

Let K(x) = K(|x|) = inf|y|=r K(y) if |x| = r (K is as in (5)) and consider the eigenvalue problem−∆pu = λK(x) |u|p−2 u in Rn,u(x) → 0 as |x| → ∞.

(18)

The existence of a positive eigenfunction ϕ1 ∈ C1,β(Rn), β ∈ (0, 1) corresponding to the first eigenvalue λ1 of (18) isestablished in [24]. It can be also verified that ϕ1 has properties (ii) and (iii) of φ1 in Lemma 3.1. Define ψ := λδϕ

γ

1 , whereδ ∈ ( 1

p−1+β ,1

p−1 ], and γ ∈ (1, (p−1)(α−p)(n−p)(p−1+β) ). Then we can prove the following result similar to Lemma 4.1:

Lemma 5.1. There exists r > 0 such that the following two inequalities hold:

(i) There exist µ > 0 such that I = (γ − 1)(p − 1) |∇ϕ1|p

ϕp−γ (p−1+β)1 K(x)

− λ1ϕγ (p−1)+γ β1 ≥ µ in Bc

r .

(ii) There exist m > 0 such that ϕ1 ≥ m in Br .

Proof of Theorem 1.4. Using Lemma 5.1, it is now easy to show that there exists a λ0 > 0 such that for λ ≥ λ0, ψ is asubsolution of (5) by following the proof of Theorem 1.3.

Next, we let e ∈ C(Rn) ∩ C1,βloc (R

n) be the weak solution of

−∆pe = K(x) in Rn.

P. Drábek, L. Sankar / Nonlinear Analysis 109 (2014) 148–155 155

The existence and decay of e follows exactly as in Lemma 3.4. Then similar to the proof in Section 4, one can show that thereexistsM(λ) ≫ 1 such that Z = M(λ)e is a supersolution of (5) for λ ≥ λ0 and satisfyψ ≤ Z . Hence by Theorem 1.2, (5) hasa positive weak solution u ∈ C1,β

loc (Rn), β ∈ (0, 1) such that ψ ≤ u ≤ Z . Clearly u(x) decays to 0 as |x| → ∞.

Acknowledgments

The first author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S. The second author wasfunded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budgetof the Czech Republic.

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