Research Article Existence for Elliptic Equation Involving Decaying … · 2019. 7. 31. · Research Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials

Research ArticleExistence for Elliptic Equation Involving DecayingCylindrical Potentials with Subcritical and Critical Exponent

Mohammed El Mokhtar Ould El Mokhtar

Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraidah 51452, Saudi Arabia

Correspondence should be addressed to Mohammed El Mokhtar Ould El Mokhtar; [email protected]

Received 4 July 2015; Accepted 13 October 2015

Academic Editor: Gershon Wolansky

Copyright © 2015 Mohammed El Mokhtar Ould El Mokhtar. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent.We will obtain a local minimizer by using Ekeland’s variational principle.

1. Introduction

In this paper, we study the existence of nontrivial solutions ofthe following problem:

− Δ𝑢 − 𝜇

𝑦

−2

𝑢 =

𝑦

−𝑎𝛾

|𝑢|

𝛾−2

𝑢 (1 + 𝜆𝑔 (𝑥))

in R𝑁, 𝑦 ̸= 0, 𝑢 > 0,(P𝜆,𝜇

)

where 𝑦 ∈ R𝑘, and let 𝑘 and 𝑁 be integers such that 𝑁 ≥ 3and 𝑘 belongs to {1, . . . , 𝑁}. 2∗ = 2𝑁/(𝑁 − 2) is the criticalSobolev exponent, 𝛾 ≤ 2∗, 0 ≤ 𝑎 < 1, 𝑔 is a continuousfunction on R𝑁, and 𝜆 and 𝜇 are parameters which we willspecify later.

We denote point 𝑥 in R𝑁 by the pair (𝑦, 𝑧) ∈ R𝑘 ×R𝑁−𝑘,D1,20

= D1,20

((R𝑘 \ {0}) × R𝑁−𝑘), andH𝜇

= H𝜇

((R𝑘 \ {0}) ×

R𝑁−𝑘), the closure of 𝐶∞0

((R𝑘 \ {0}) × R𝑁−𝑘) with respect tothe norms

‖𝑢‖ = (∫

R𝑁|∇𝑢|

2

)

1/2

,

‖𝑢‖

𝜇

= (∫

R𝑁(|∇𝑢|

2

− 𝜇

𝑦

−2

|𝑢|

2

) 𝑑𝑥)

1/2

(1)

with 𝜇 < 𝜇𝑘

= ((𝑘 − 2)/2)

2 for 𝑘 ̸= 2.From the Hardy inequality, it is easy to see that the norm

‖𝑢‖

𝜇

is equivalent to ‖𝑢‖.

We define the weighted Sobolev space D := H𝜇

∩

𝐿

𝛾

(R𝑁, |𝑦|−𝑏𝑑𝑥) ∩ 𝐿2(R𝑁, |𝑦|−2𝑑𝑥) with 𝑏 = 𝑎𝛾, which is aBanach space with respect to the norm defined by N(𝑢) :=‖𝑢‖

𝜇

+ (∫

R𝑁|𝑦|

−𝑏

|𝑢|

𝛾

𝑑𝑥)

1/𝛾.Mymotivation of this study is the fact that such equations

arise in the search for solitary waves of nonlinear evolutionequations of the Schrödinger or Klein-Gordon type (cf.[1–3]). Roughly speaking, a solitary wave is a nonsingularsolution which travels as a localized packet in such a way thatthe physical quantities corresponding to the invariances ofthe equation are finite and conserved in time. Accordingly, asolitary wave preserves intrinsic properties of particles suchas the energy, the angular momentum, and the charge, whosefiniteness is strictly related to the finiteness of the 𝐿2-norm.Owing to their particle-like behavior, solitary waves can beregarded as a model for extended particles and they arise inmany problems ofmathematical physics, such as classical andquantum field theory, nonlinear optics, fluid mechanics, andplasma physics (see, e.g., [4]).

Several existence and nonexistence results are availablein the case 𝑘 = 𝑁, and we quote, for example, [5–7] andthe references therein. When 𝜇 = 0, 𝑔(𝑥) ≡ 1; problem(P𝜆,𝜇

) has been studied in the famous papers by Brézis andNirenberg [8] and Xuan [9] which consider the existence andnonexistence of nontrivial solutions to quasilinear Brézis-Nirenberg-type problems with singular weights.

Concerning the existence result in the case 𝑘 < 𝑁, wecite [10, 11] and the references therein. As noticed in [10], for

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2015, Article ID 494907, 4 pageshttp://dx.doi.org/10.1155/2015/494907

2 International Journal of Differential Equations

𝜇 < 0 and 𝑎 = 0, Badiale and Rolando have considered theproblem (P

0,𝜇

). They established the existence of nontrivialnonnegative radial solution when 𝛽 ∈ (0, 2) and 𝛾 ∈ (2

𝛽

, 2

∗

)

or 𝛽 ∈ (2, +∞) and 𝛾 ∈ (2∗, 2𝛽

); in addition, if the function𝑓(𝑢) = |𝑢|

𝛾−1

𝑢 is odd, then (P0,𝜇

) has infinitely many radialsolutions. In [5], Badiale et al. proved the nonexistence ofnonzero classical solutions when 𝑘 ≤ 𝑁 and the pair (𝛽, 𝛾)belongs to the light gray region. That is, (𝛽, 𝛾) ∈ A = A

1

∪

A2

∪A3

, where

A1

:= {(𝛽, 𝛾) ∈R2

: 𝛽 ∈ (0, 2) , 𝛾 ∉ (2

𝛽

, 2

∗

) , 𝛾 ≥ 2} \ {(2, 2

∗

)} ,

A2

:= {(𝛽, 𝛾) ∈R2

: 𝛽 ∈ (2,𝑁) , 𝛾 ∉ (2

∗

, 2

𝛽

) , 𝛾 ≥ 2} ,

A3

:= {(𝛽, 𝛾) ∈R2

: 𝛽 ∈ [𝑁, +∞) , 𝛾 ∈ [2, 2

∗

]} .

(2)

Since our approach is variational, we define the functional𝐼

𝜆,𝜇

onD by

𝐼

𝜆,𝜇

(𝑢) := (

1

2

) ‖𝑢‖

2

𝜇

− (

1

𝛾

)∫

R𝑁

𝑦

−𝑏

|𝑢|

𝛾

(1 + 𝜆𝑔 (𝑥)) 𝑑𝑥.

(3)

We say that 𝑢 ∈ D is a weak solution of the problem (P𝜆,𝜇

) ifit is a nontrivial nonnegative function and satisfies

⟨𝐼

𝜆,𝜇

(𝑢) , V⟩ := ∫R𝑁

(∇𝑢∇V − 𝜇

𝑦

−2

𝑢V

−

𝑦

−𝑏

|𝑢|

𝛾−2

𝑢V (1 + 𝜆𝑔 (𝑥))) = 0, for V ∈ D.(4)

Throughout this work, we consider the following regionsR1

,R2

, such that

R1

:= {(2, 𝛾) ∈R2

: 𝛾 ∈ (2

2−2𝑎

, 2

∗

)} ,

R2

:= {(2, 𝛾) ∈R2

: 𝛾 ∈ (2, 2

2−2𝑎

)}

(5)

with 22−2𝑎

= 2𝑁/(𝑁 − (2 − 2𝑎)).Concerning the perturbation 𝑔, we assume

𝑔 ∈ 𝐿

∞

(R𝑁

) ,

𝑔 (𝑥) > 0 ∀𝑥 ∈ R𝑁

.

(𝐺)

In our work, we prove the existence of at least one criticalpoint of 𝐼

𝜆,𝜇

by Ekeland’s variational principle in [12].We will state our main result.

Theorem 1. Assume that 2 < 𝑘 ≤ 𝑁, 𝜇 < 𝜇𝑘

, 0 < 𝑎 < 1, and(𝐺) hold.

If (2, 𝛾) ∈ R1

∪ R2

, then there exists Λ∗ > 0 such thatthe problem (P

𝜆,𝜇

) has at least one nontrivial solution for any𝜆 > Λ

∗.

This paper is organized as follows. In Section 2, we givesome preliminaries. Section 3 is devoted to the proof ofTheorem 1.

2. Preliminaries

We list here a few integrals inequalities. The first inequalitythat we need is the weighted Hardy inequality [13]

𝜇

𝑘

∫

R𝑁

𝑦

−2 V2𝑑𝑥 ≤ ∫R𝑁

|∇V|2 𝑑𝑥, ∀V ∈ H𝜇

. (6)

The starting point for studying (P𝜆,𝜇

) is the Hardy-Sobolev-Maz’ya inequality that is peculiar to the cylindrical case𝑘 < 𝑁and that was proved by Gazzini and Musina in [14]. It statesthat there exists positive constant 𝐶

𝛾

such that

𝐶

𝛾

(∫

R𝑁|V|𝛾 𝑑𝑥)

2/𝛾

≤ ∫

R𝑁(|∇V|2 − 𝜇

𝑦

−2 V2) 𝑑𝑥, (7)

for 𝜇 = 0; equation of (P𝜆,𝜇

) is related to a family ofinequalities given byCaffarelli et al. [15], for any V ∈ 𝐶∞

𝑐

((R𝑘\

{0}) × R𝑁−𝑘). The embedding H𝜇

→ 𝐿

𝛾

(R𝑁, |𝑦|−𝑏𝑑𝑥) iscompact, where 𝑏 = 𝑎𝛾 and 𝐿𝛾(R𝑁, |𝑦|−𝑏𝑑𝑥) is the weighted𝐿

𝛾 space with respect to the norm

|𝑢|

2

𝛾,𝑏

= (∫

R𝑁

𝑦

−𝑏

|V|𝛾 𝑑𝑥)2/𝛾

.

(8)

Definition 2. Assume 2 ≤ 𝑘 < 𝑁, 0 < 𝜇 ≤ 𝜇𝑘

, and 2 < 𝛾 < 2∗.Then, the infimum 𝑆

𝜇,𝛾

defined by

𝑆

𝜇,𝛾

= 𝑆

𝜇,𝛾

(𝑘, 𝛾) := infV∈D\{0}

∫

R𝑁(|∇V|2 − 𝜇

𝑦

−2 V2) 𝑑𝑥

(∫

R𝑁

𝑦

−𝑏

|V|𝛾 𝑑𝑥)2/𝛾

(9)

is achieved onH𝜇

.

Lemma 3. Let (𝑢𝑛

) ⊂ D be a Palais-Smale sequence ((𝑃𝑆)𝛿

for short) of 𝐼𝜆,𝜇

such that

𝐼

𝜆,𝜇

(𝑢

𝑛

) → 𝛿,

𝐼

𝛽,𝜆,𝜇

(𝑢

𝑛

) → 0

𝑖𝑛 D

(𝑑𝑢𝑎𝑙 𝑜𝑓 D) 𝑎𝑠 𝑛 → ∞,

(10)

for some 𝛿 ∈ R. Then, 𝑢𝑛

⇀ 𝑢 inD and 𝐼𝛽,𝜆,𝜇

(𝑢) = 0.

Proof. From (10), we have

(

1

2

)

𝑢

𝑛

2

𝜇

− (

1

𝛾

)∫

R𝑁

𝑦

−𝑏

𝑢

𝑛

𝛾

(1 + 𝜆𝑔 (𝑥)) 𝑑𝑥

= 𝛿 + 𝑜

𝑛

(1) ,

𝑢

𝑛

2

𝜇

− ∫

R𝑁

𝑦

−𝑏

𝑢

𝑛

𝛾

(1 + 𝜆𝑔 (𝑥)) 𝑑𝑥 = 𝑜

𝑛

(1) ,

for 𝑛 large,

(11)

where 𝑜𝑛

(1) denotes 𝑜𝑛

(1) → 0 as 𝑛 → ∞. Then,

𝛿 + 𝑜

𝑛

(1) = 𝐼

𝜆,𝜇

(𝑢

𝑛

) − (

1

𝛾

)⟨𝐼

𝛽,𝜆,𝜇

(𝑢

𝑛

) , 𝑢

𝑛

⟩

= (

(𝛾 − 2)

2𝛾

)

𝑢

𝑛

2

𝜇

,

(12)

International Journal of Differential Equations 3

and (𝑢𝑛

) is bounded in D. Going if necessary to a subse-quence, we can assume that there exists 𝑢 ∈ D such that

𝑢

𝑛

⇀ 𝑢 in D,

𝑢

𝑛

→ 𝑢 in 𝐿𝛾 (R𝑁,

𝑦

−𝑏

𝑑𝑥) ,

𝑢

𝑛

→ 𝑢 a.e. in R𝑁.

(13)

Consequently, we get, for all V ∈ 𝐶∞0

((R𝑘 \ {0}) ×R𝑁−𝑘),

∫

R𝑁(∇𝑢∇V − 𝜇

𝑦

−2

𝑢V

−

𝑦

−𝑏

|𝑢|

𝛾−2

𝑢V (1 + 𝜆𝑔 (𝑥))) = 0,(14)

which means that

𝐼

𝛽,𝜆,𝜇

(𝑢) = 0. (15)

3. Existence Result

Firstly, we require the following lemmas.

Lemma 4. Let (𝑢𝑛

) ⊂ D be a (𝑃𝑆)𝛿

sequence of 𝐼𝜆,𝜇

for some𝛿 ∈ R. Then,

𝑢

𝑛

⇀ 𝑢 𝑖𝑛 D (16)

and either

𝑢

𝑛

→ 𝑢

𝑜𝑟 𝛿 ≥ 𝐼

𝜆,𝜇

(𝑢) + (

(𝛾 − 2)

2𝛾

) (𝑆

𝜇,𝛾

)

𝛾/(𝛾−2)

.

(17)

Proof. We know that (𝑢𝑛

) is bounded in D. Up to a subse-quence if necessary, we have that

𝑢

𝑛

⇀ 𝑢 in D

𝑢

𝑛

→ 𝑢 a.e. in R𝑁.(18)

Denote V𝑛

= 𝑢

𝑛

− 𝑢, and then V𝑛

⇀ 0. As in Brézis and Lieb[16], we have

∫

R𝑁

𝑦

−𝑏

𝑢

𝑛

𝛾

= ∫

R𝑁

𝑦

−𝑏

V𝑛

𝛾

+ ∫

R𝑁

𝑦

−𝑏

|𝑢|

𝛾

,

𝑢

𝑛

2

𝜇

=

V𝑛

2

𝜇

+ ‖𝑢‖

2

𝜇

.

(19)

From Lebesgue theorem and by using the assumption (𝐺), weobtain

lim𝑛→∞

∫

R𝑁𝑔 (𝑥)

𝑦

−𝑏

𝑢

𝑛

𝛾

𝑑𝑥

= lim𝑛→∞

∫

R𝑁𝑔 (𝑥)

𝑦

−𝑏

|𝑢|

𝛾

𝑑𝑥.

(20)

Then, we deduce that

𝐼

𝜆,𝜇

(𝑢

𝑛

) = 𝐼

𝜆,𝜇

(𝑢) + (

1

2

)

V𝑛

2

𝜇

− (

1

𝛾

)∫

R𝑁

𝑦

−𝑏

V𝑛

𝛾

+ 𝑜

𝑛

(1) ,

⟨𝐼

𝜆,𝜇

(𝑢

𝑛

) , 𝑢

𝑛

⟩ =

V𝑛

2

𝜇

− ∫

R𝑁

𝑦

−𝑏

V𝑛

𝛾

+ 𝑜

𝑛

(1) .

(21)

From the fact that V𝑛

⇀ 0 inD, we can assume that

lim𝑛→∞

V𝑛

2

𝜇

= lim𝑛→∞

∫

R𝑁

𝑦

−𝑏

V𝑛

𝛾

= 𝛼 ≥ 0. (22)

Assuming that 𝛼 > 0, we have by definition of 𝑆𝜇,𝛾

𝛼 ≥ 𝑆

𝜇,𝛾

𝑙

(2/𝛾)

, (23)

and so

𝛼 ≥ (𝑆

𝜇,𝛾

)

𝛾/(𝛾−2)

.

(24)

Then, we get

𝛿 ≥ 𝐼

𝜆,𝜇

(𝑢) + (

(𝛾 − 2)

2𝛾

) (𝑆

𝜇,𝛾

)

𝛾/(𝛾−2)

. (25)

Therefore, if not, we obtain 𝛼 = 0. That is, 𝑢𝑛

→ 𝑢 inD.

Lemma 5. Suppose that 2 < 𝑘 ≤ 𝑁, 𝜇 < 𝜇𝑘

, and (𝐺) hold. If(2, 𝛾) ∈ R

1

∪R2

, then there exist Λ∗ > 0 and and ] positiveconstants such that, for all 𝜆 > Λ∗,

(i) there exist 𝜔 ∈ R𝑁 such that 𝐼𝜆,𝜇

(𝜔) < 0,

(ii) we have

𝐼

𝜆,𝜇

(𝑢) ≥ ] > 0 𝑓𝑜𝑟 ‖𝑢‖𝜇

=

0

. (26)

Proof. (i) Let 𝑡0

> 0 where 𝑡0

is small, and 𝜙 ∈ 𝐶∞0

((R𝑘 \

{0}) × R𝑁−𝑘) such that 𝜙 ̸≡ 0. Choosing Λ∗ = |𝑡0

𝜙|

1−𝛾, then,if 𝜆 > Λ∗ large enough,

𝐼

𝜆,𝜇

(𝑡

0

𝜙) := (

𝑡

2

0

2

)

𝜙

2

𝜇

− (

𝑡

𝛾

0

𝛾

)∫

R𝑁

𝑦

−𝑏

𝜙

𝛾

1

− (

𝑡

𝛾

0

𝛾

)∫

R𝑁

𝑦

−𝑏

𝜙

𝛾

𝜆𝑔 (𝑥)

< (

𝑡

2

0

2

)

𝜙

2

𝜇

− (

𝑡

𝛾

0

𝛾

)∫

R𝑁

𝑦

−𝑏

𝜙

𝛾

1

− (

𝑡

0

𝛾

)∫

R𝑁

𝑦

−𝑏

𝜙

𝑔 (𝑥) < 0.

(27)

Thus, if 𝜔 = 𝑡0

𝜙, we obtain that 𝐼𝜆,𝜇

(𝜔) < 0.

4 International Journal of Differential Equations

(ii) By the Holder inequality and the definition of 𝑆𝜇,𝛾

andsince 𝛾 > 2, we get for all 𝑢 ∈ D \ {0}

𝐼

𝜆,𝜇

(𝑢) := (

1

2

) ‖𝑢‖

2

𝜇

− (

1

𝛾

)∫

R𝑁

𝑦

−𝑏

|𝑢|

𝛾

(1 + 𝜆𝑔 (𝑥)) 𝑑𝑥

≥ (

1

2

) ‖𝑢‖

2

𝜇

− (

1

𝛾

) 𝑆

𝜇,𝛾

‖𝑢‖

𝛾

𝜇

(1 + 𝜆

𝑔

∞

) .

(28)

If 𝜆 > Λ∗, then there exist ] > 0 and 0

> 0 small enoughsuch that

𝐼

𝜆,𝜇

(𝑢) ≥ ] > 0 for ‖𝑢‖𝜇

=

0

. (29)

We also assume that 𝑡0

is small enough such that ‖𝑡0

𝜙‖

𝜇

<

0

.Thus, we have

𝑐

1

= inf {𝐼𝜆,𝜇

(𝑢) : 𝑢 ∈ 𝐵

0} < 0,

where 𝐵0

= {𝑢 ∈ D, N (𝑢) ≤ 0

} .

(30)

Using Ekeland’s variational principle, for the complete metricspace 𝐵

𝜌0with respect to the norm of D, we can prove that

there exists a (𝑃𝐶)𝑐1sequence (𝑢

𝑛

) ⊂ 𝐵

𝜌0such that 𝑢

𝑛

⇀ 𝑢

1

for some 𝑢1

withN(𝑢1

) ≤ 𝜌

0

.Now, we claim that 𝑢

𝑛

→ 𝑢

1

. If not, by Lemma 4, we have

𝑐

1

≥ 𝐼

𝜆,𝜇

(𝑢

1

) + (

(𝛾 − 2)

2𝛾

) (𝑆

𝜇,𝛾

)

𝛾/(𝛾−2)

≥ 𝑐

1

+ (

(𝛾 − 2)

2𝛾

) (𝑆

𝜇,𝛾

)

𝛾/(𝛾−2)

> 𝑐

1

,

(31)

which is a contradiction.Then, we obtain a critical point 𝑢

1

of 𝐼𝜆,𝜇

for all 𝜆 > Λ∗large enough satisfying

𝑐

1

= (

(𝛾 − 2)

2𝛾

)

𝑢

1

2

𝜇

> 0. (32)

Proof of Theorem 1. From Lemmas 4 and 5, we can deducethat there exists at least a nontrivial solution 𝑢

1

for ourproblem (P

𝜆,𝜇

) with positive energy.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

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