On an Anisotropic Equation with Critical Exponent and Non-Standard Growth Condition

CHUNG Nguyen Thanh

Department of Science Management and International Cooperation,Quang Binh University,312 Ly Thuong Kiet,Dong Hoi,Quang Binh,Vietnam.

Received 4 July 2012;Accepted 2 April 2013

On an Anisotropic Equation with Critical Exponent and Non-Standard Growth Condition

CHUNG Nguyen Thanh?

Department of Science Management and International Cooperation,Quang Binh University,312 Ly Thuong Kiet,Dong Hoi,Quang Binh,Vietnam.

Received 4 July 2012;Accepted 2 April 2013

.Using variational methods,we prove the existence of a nontrivial weak solution for the problem

AMS Subject Classifications:35D05,35J60,35J70,58E05,68T40

Chinese Library Classifications:O175.25

Anisotropic equation;critical exponent;variational methods;existence.

1 Introduction and preliminaries

In this paper,we are interested in the existence of a nontrivial weak solution for the problem

is the critical exponent for this class of problem,and λ is a parameter.

In the case when pi(x)=p(x)for any i=1,2,···,N,the operator involved in(1.1) has similar properties to the p(x)-Laplace operator,i.e.,Δp(x)u:=div(|?u|p(x)?2?u). This differential operator is a natural generalization of the isotropic p-Laplace operator Δpu:=div(|?u|p?2?u),where p＞1 is a real constant.However,the p(x)-Laplace operator possesses more complicated nonlinearities than the p-Laplace operator,due to the fact that Δp(x)is not homogeneous.The study of nonlinear elliptic problems(equations and systems)involving quasilinear homogeneous type operators like the p-Laplace operator is based on the theory of standard Sobolev spaces Wk,p(?)in order to find weak solutions.These spaces consist of functions that have weak derivatives and satisfy certain integrability conditions.In the case of nonhomogeneous p(x)-Laplace operators the natural setting for this approach is the use of the variable exponent Sobolev spaces.Differential and partial differential equations with non-standard growth conditions have received specific attention in recent decades.The interest played by such growth conditions in elastic mechanics and electrorheological fluid dynamics has been highlighted in many physical and mathematical works.

In a recent paper[1],I.Fragal`a et al.have studied the following anisotropic quasilinear elliptic problem

Motivated by the above papers,the goal of this note is to show the existence of at least one nontrivial weak solution for problem(1.1)with the critical exponent

Throughout this paper,we assume that

The main result of this work can be described as follows.

It should be noticed that problem(1.1)with Laplace operator?Δpu was studied by L.Calota?[3].So,our paper is a natural extension from[3]to the anisotropic case.

In order to study problem(1.1),we will appeal to the variable exponent Lebesgue spaces Lq(x)(?).In that context,we refer to the book of Musielak[4],the papers of Kova′cˇik and Ra′kosn′?k[5],Fan et al.[6,7]and Miha?ilescu et al.[8].Set

We recall the following so-called Luxemburg norm on this space defined by the formula

For any u∈Lp(x)(?)and v∈Lp′(x)(?)the H¨older inequality

holds true.

Animportantroleinmanipulating thegeneralizedLebesgue-Sobolevspacesisplayed by the modular of the Lp(x)(?)space,which is the mapping ρp(x):Lp(x)(?)→R defined by

If u∈Lp(x)(?)and p+＜∞then the following relations hold

provided|u|p(x)＞1 while

provided|u|p(x)＜1 and

2 Proof of the main result

Then,a simple computation shows that Jλ∈C1(X,R)and

for all v∈X.Thus,weak solutions of problem(1.1)are exactly the critical points of the functional Jλ.

Lemma 2.1.There exists λ?＞0 such that for any λ∈(0,λ?),there exist two constant ρ,α＞0 such that Jλ(u)≥α for all u∈X with‖u‖X=ρ.

Proof.Observe that

From(2.2)-(2.4),we get

where C3is a positive constant.

On the other hand,we focus our attention on the case when u∈X with‖u‖X＜1.For such element u,we have|?xiu|pi＜1 for any i=1,2,···,N.Hence,

By the definition of the Jλ,we have

Define the function

for t∈(0,∞).

Lemma 2.2.For any λ∈(0,λ?)as in Lemma 2.1,there exists ?∈X such that ?≥0,?6=0 and Jλ(t?)＜0 for all t＞0 small enough.

Proof.By(1.4),there exists a constant δ∈(1,p?),such that the set ?0:={x∈?:q(x)＜δ} is nonempty and|?0|＞0.

It is clear that Jλ(t?)＜0,provided that

and the Lemma 2.2 is proved.

Proof Theorem 1.1.By(2.6),the functional Jλis bounded from below onρ(0).Applying the Ekeland variational principle in[9]to the functional Jλ:ρ(0)→R,it follows that there exists u?∈ρ(0)such that

By Lemmas 2.1 and 2.2,we have

Let us choose ?＞0 such that

for all t＞0 small enough and all v∈Bρ(0).The above information shows that

Therefore,there exists a sequence{um}?Bρ(0)such that

where X?is the dual space of X.

Next,we shall prove that

for all v∈X.This follows that u is a weak solution of problem(1.1).

We now prove that u6=0.Indeed,assume by contradiction that u≡0 and

By(2.8),we have

or

It implies that

Using again(2.8),from(1.4)and(2.11),we have

which is a contradiction.We conclude that u6=0.The proof of Theorem 1.1 is completely proved.

Acknowledgments

The author would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

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?Corresponding author.Email address:ntchung82@yahoo.com(N.T.Chung)

10.4208/jpde.v26.n3.2 September 2013