Zheng Zhouand Xin Si

1School of Applied Mathematical,Xiamen University of Technology,Xiamen,Fujian 361024,P.R.China.

Infinitely Many Clark Type Solutions to a p(x)-Laplace Equation∗

Zheng Zhou1,∗and Xin Si2

1School of Applied Mathematical,Xiamen University of Technology,Xiamen,Fujian 361024,P.R.China.

.In this paper,the followingp(x)-Laplacian equation:is studied.By applying an extension of Clark’s theorem,the existence of in finitely many solutions as well as the structure of the set of critical points near the origin are obtained.

AMS subject classifications:35A15,35A05,35J70

Chinese Library Classifications:O175.29,O176.3

Clark theorem,in finitely many solutions,p(x)-Laplace,variational methods.

1 Introduction

The Clark theorem[2]is an important tool in critical point theory,which is constantly and effectively applied to sublinear differential equations with symmetry.A variant of the Clark Theorem was given by Heinz in[8].

Theorem 1.1.Let X be a Banach space,Φ∈C1(X,R).Assume thatΦsatisfies the(PS)condition,is even and bounded from below,andΦ(0)=0.If for any k∈N,there exists a k-dimensional subsequence Xkof X andρk＞0such thatsupXk∩SρkΦ＜0,where Sρ={u∈X|‖u‖=ρ},thenΦhas a sequence of critical values ck＜0satisfying ck→0as k→∞.

Theorem1.1 asserts the existence of a sequence of critical valuesck＜0 satisfyingck→0 ask→∞,without giving any information on the structure of the set of critical points.Avery interesting question arising from Theorem 1.1 is whether there are a sequence of critical pointsuksuch that Φ(uk)→0 and ‖uk‖→0 ask→∞ under the assumptions of Theorem 1.1.In[11],the authors answered this question and gave the structure of the set of critical points near the original in the abstract setting of Clark’s theorem.One of the results is the following.

Theorem 1.2.Let X be a Banach space,Φ∈C1(X,R).Assume thatΦsatisfies the(PS)condition,is even and bounded from below,andΦ(0)=0.If for any k∈N,there exists a k-dimensional subsequence Xkof X andρk＞0such thatsupXk∩SρkΦ＜0,where Sρ={u∈X|‖u‖=ρ},then at least one of the following conclusions holds.

(i)There exist a sequence of critical points uksatisfyingΦ(uk)＜0for all k and‖uk‖→0as k→∞.

(ii)There exists r＞0such that for any0＜a＜r,there exists a critical point u such that‖u‖=a andΦ(u)=0.

In[11],the authors got some variants of Clark Theorem which were applied to indefinite problems such as problems on periodic solutions of first order Hamiltonian systems.And the Theorem 1.2 is applied to ap-Laplace equation on RN.i.e.,

wherep＞1.Assuming

(a1)there exists δ＞0,1≤γ＜p,C＞0 such thatf∈C(RN×[−δ,δ],R),fis odd inu,|f(x,u)|≤C|u|γ−1,and limu→0F(x,u)/|u|p=+∞ uniformly in some ballBr(x0)⊂RN;

(a2)V,Q∈C(RN,R1),V(x)≥α0and 0＜Q(x)≤β0for some α0＞0,β0＞0,andM≜

With conditions(a1)and(a2),equation(1.1)has in finitely many solutionsuksuch that‖uk‖L∞→0 ask→∞.

In this paper,the followingp(x)-Laplacian equation(p(x)＞1)is considered:

where Δp(x)u=div(|∇u|p(x)−2∇u),p(x)∈C(RN).

Whenp(x)≡const.,problem(1.2)is the equation(1.1),which was studied in[11].The variable exponent Sobolev spaceW1,p(x)(Ω)is a natural generalization of the classical Sobolev spaceW1,p(Ω).The variable exponentp(x)-Laplacian equations arise from nonlinear elastic mechanics(see[15])and electrorheological fluids(see[1,12]).And thep(x)-Laplacian operator possesses more complicated nonlinearities,for example,it is inhomogeneous,so in the discussion,some special techniques needed will be given in Section 2.

Following the argument in[11]and using Theorem 1.2,we get the following result.

Theorem 1.3.Assume

(b1)there existsδ＞0,1≤ γ＜p−,C＞0such that f∈C(RN×[−δ,δ],R),f is odd in u,|f(x,u)|≤C|u|γ−1,andlimu→0F(x,u)/|u|p(x)=+∞uniformly in some ball Br(x0)⊂RN,

(b2)V,Q∈C(RN,R1),V(x)≥α0and0＜Q(x)≤ β0for someα0＞0,β0＞0,and M≜

Then equation(1.2)has in finitely many solutions uksuch that‖uk‖L∞→0as k→∞.

2 Preliminaries

Let Ω⊂RN(N≥2),p(x)∈C(Ω),set

The spaceLp(x)(Ω)is a Banach space endowed with the Luxemburg norm

Proposition 2.1.(see[4])

(2)Ifp1,p2∈C+(¯Ω),p1(x)≤p2(x)for anyx∈¯Ω,thenLp2(x)(Ω)■→Lp1(x)(Ω)and the embedding is continuous.

Proposition 2.2.(see[4,9])Set ρ(u)=RΩ|u(x)|p(x)dx.Ifu,uk∈Lp(x)(Ω),we have

(1)foru/=0,‖u‖p(x)=λ⇔ρ(uλ)=1;

(2)‖u‖p(x)＜1(=1;＞1)⇔ρ(u)＜1(=1,＞1);

The spaceW1,p(x)(Ω)is defined by

and equipped with the norm

We denote the closure ofC∞0(Ω)inW1,p(x)(Ω)byW1,p(x)0(Ω)and

Proposition 2.3.(see[4])

(3)There is a constantC＞0 such that

Proposition 2.4.(see[5])De fineI(u)=R(|∇u|p(x)+a(x)|u|p(x)),where 0＜a−＜a(x)＜∞,use the norm

Letu∈W1,p(x)(Ω),then

(i)‖u‖＜1(=1,＞1)⇔I(u)＜1(=1,＞1);

(ii)if‖u‖≥1,then ‖u‖p−≤I(u)≤‖u‖p+;

(iii)if‖u‖≤1,then ‖u‖p+≤I(u)≤‖u‖p−;

(iv)I(un)→0⇔‖un‖→0;

(v)I(un)→∞⇔‖un‖→∞.

Definition 2.1.u∈W1,p(x)(RN)is called a weak solution of problem(1.2)if

and

We know that the critical points of Φ are just the weak solutions of problem(1.2).

3 Proof of Theorem 1.1

Proof.We prove it in three steps:

Step 1.Construct proper functional and get the coerciveness.Chooseˆf∈C(RN×R,R)so thatˆfis odd inu∈R,ˆf(x,u)=f(x,u)forx∈R and|u|＜δ/2,andˆf(x,u)=0 forx∈RNand|u|＞δ.In order to obtain solutions of(1.2)we study

which is the Euler equation of the functional

whereXis the Banach space

endowed with the norm(see[5])

Therefore,

and then Φ is coercive and bounded below.

Step 2.The(PS)condition holds.Let{un}be a(PS)sequence,thus Φ(un)is bounded and Φ′(un)→0.Then{un}is bounded.Assumeun⇀uinW1,p(x)(RN)and strongly inThen Φ′(u)→0,and

wherec＞0 is a constant.

Forp(x)≥2,it is easy to see that

In the case 1＜p(x)＜2,for anyv,w∈Lp(x)(RN),we have

Thus

Substitutew=|un−u|,v=|un|+|u|to(3.8),(3.10),we have for 1＜p(x)＜2,

Now we estimateI2,for anyR＞0,

where we used H¨older inequality(Proposition 2.1)in(3.12),Proposition 2.2 in(3.13),and the definition ofXin(3.14).By the condition(b2),we have‖M‖L1(RNBR(0))→0 asR→∞.Lp(x)■→Lγleads to ‖un−u‖Lγ(BR(0))→0.Thus

Combining with(3.7),(3.9),(3.11)and(3.15),{un}converges strongly inXand the(PS)condition holds for Φ.

Step 3.Equation(1.2)has in finitely many Clark type solutions.For anyK＞0,there exists δ=δ(K)＞0 such that ifu∈C∞0(Br(x0))and|u|∞＜δ,thenQ(x)ˆF(x,u)≥K|u(x)|p(x),and thus

This implies,for anyk∈N,ifXkis ak-dimensional subsequence ofC∞0(Br(x0))and ρk＞0 is sufficiently small,then supXk∩SρkΦ＜0,whereSρ={u∈X:‖u‖=ρ}.Now we appeal to Theorem 1.2 to obtain in finitely many solutions{uk}for(3.1)such that‖uk‖→0 ask→∞.

In the following,we show that‖uk‖L∞→0 ask→∞.In the case 1＜p(x)＜N,denote.Letube a solution of(3.1)and constant α＞0.LetT＞0 and setuT(x)=max{−T,min{u(x),T}}.Multiplying both sides of(3.1)with|uT|αuTimplies

Since

and

Combining(3.16)-(3.18)with Sobolev inequality,we have

Thus

whereC≥ 1,independent ofuand α.Setthat is···.From(3.21),an iterating process leads to

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19 August 2014;Accepted 29 November 2014

∗Supported by National Natural Science Foundation of China(No.11326150)and Foundation of Fujian Educational Committee(JA11240)

∗Corresponding author.Email addresses:zhouzhengslx@163.com(Z.Zhou),xsi@xmut.edu.cn(X.Si)