Department of Mathematics,Razi University,Kermanshah,Iran E-mail:nyamoradi@razi.ac.ir;neamat80@yahoo.com

Abdolrahman RAZANI

Department of Pure Mathematics,Faculty of Science,Imam Khomeini International University,34149-16818,Qazvin,Iran E-mail:razani@sci.ikiu.ac.ir

Abstract In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:whereis the fractional p-Laplacian with 0ps,a,b>0,λ>0 is a parameter,V:RN→R+is a potential function,θ=is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We get the existence of in fi nitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory.To the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.

Key words Hardy-Little wood-Sobolev inequality;concentration-compactness principle;variational method;Fractional p-Laplacian operators;multiple solutions

1 Introduction

In this paper,we study the multiplicity of solutions to the following Kirchhoff-type equations with Hardy-Little wood-Sobolev critical nonlinearity:

So,the main result of this paper can be included in the following theorem:

Theorem 1.1Assume that(V1)and(V2)hold.Then,(1.1)has in finitely many solutions forsp

We point out that Theorem 1.1 extends Theorem 2.4 in[21]to the casea=1,b=0,p=2 and Theorem 1.1 in[14,17,32]to the casep=2 ands=1.

We firstly recall that the fractional Kirchhoff equation was first introduced and motivated in[6].The study of existence and uniqueness of positive solutions for Choquard type equations attracted a lot of attention of researchers due to its vast applications in physical models[26].Fractional Choquard equations and their applications is very interesting;we refer the readers to[3,7,8,13,18,21,22,27,35,37]and the references therein.The authors in[27],by using the Mountain Pass Theorem and the Ekeland variational principle,obtained the existence of nonnegative solutions of a Schrdinger-Choquard-Kirchhoff-type fractionalp-equation.Ma and Zhang[18]studied the fractional order Choquard equation and proved the existence and multiplicity of weak solutions.In[7],the authors investigated a class of Brzis-Nirenberg type problems of nonlinear Choquard equation involving the fractional Laplacian in bounded domain?.Wang and Yang[36],by using an abstract critical point theorem based on a pseudo-index related to the cohomological index,studied the bifurcation results for the critical Choquard problems involving fractionalp-Laplacian operator:

where ? is a bounded domain in RNwith Lipschitz boundary andλis a real parameter.Also,in[11,23,29],the authors studied the existence of multiple solutions for problem(1.3),whenp=2.For more works on the Brezis-Nirenberg type results on semilinear elliptic equations with fractional Laplacian,we refer to[29,30,33]and references therein.

On the other hand,Song and Shi in[31]studied the following Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity:

The proofs of our main results are obtained by applying variational arguments inspired by[25,31].

The paper is organized into three sections.In Section 2,we recall some basic de finitions of fractional Sobolev space and Hardy-Littlewood-Sobolev Inequality,and we give some useful auxiliary lemmas.In Section 3,we give the proof of Theorem 1.1.

2 Preliminary Lemmas

Let 0

equipped with the norm

De fine the space

with the norm

Lemma 2.1(see[25,Lemma 1])(Xλ,‖·‖Xλ)is a uniformly convex Banach space.

Lemma 2.2(see[25,Lemma 2])Assume that(V1)holds.Then the embeddingsXλWs,p(RN)Lν(RN)are continuous forν∈[].In particular,there exists a constantCν>0 such that

Moreover,for anyR>0 andν∈[1]the embeddingXλLν(BR(0))is compact.

Also,letSp,Hbe the best constant

andS?be the best Sobolev constant

Now,we recall that a sequence{(un,vn)}is a Palais-Smale sequence at the levelc((PS)csequence in short)for the functionalJifJ(un,vn)→candJ′(un,vn)→0.If any(PS)csequence{(un,vn)}has a convergent subsequence,we say thatJsatis fies the(PS)ccondition.

Lemma 2.5Assume that(V1)and(V2)hold.ThenJλ,Vsatis fies the(PS)ccondition for allsp

So,{un}nis bounded inXλ.

Then,there exists a subsequence,still denote by{un},such thatun?uweakly inXλ.Also,in view of Lemma 2.2 and Lemma 2.3,we have

So,(2.16)and(2.17)imply that

Here,similar to the method in[25],we prove that{un}converges strongly touinXλ.To this end,let?∈Xλbe fixed and denote byB?the linear functional onXλde fined by

for allv∈Xλ.By the Hlder inequality and de finition ofB?,we have

To prove our main result,we will use the Krasnoselskii’s genus introduced by Krasnoselskii in[38].LetXbe a Banach space and let us denote by Λ the class of all closed subsetsA?X{0}that are symmetric with respect to the origin;that is,u∈Aimplies?u∈A.

Theorem 2.6(See[28])LetXbe an in finite dimensional Banach space and letJ∈C1(X,R)be even functional withJ(0)=0.IfX=Y⊕Z,whereYis finite dimensional andJsatisfiesthat

(I1)There exist constantsρ,α>0 such thatJ(u)≥αfor allu∈?Bρ(0)∩Z;

(I2)There exists Θ>0 such thatJsatis fies the(PS)ccondition for 0

whereγ(V)is Krasnoselskii’s genus ofV.Forj∈N,set

So,0≤cj≤cj+1forj>kandcj<Θ,then we get thatcjis the critical value ofJ.Furthermore,ifcj=cj+1=···=cj+m=c<Θ forj>k,thenγ(Kc)≥m+1,where

Proof of Theorem 1.1We shall apply Theorem 2.6 toJλ,V.We know thatXλis a Banach space andJλ,V∈C1(Xλ,R).By(2.8),functionalJλ,Vsatis fiesJλ,V(0)=0.We divide the proof into the following four steps:

Step 1We will show thatJλ,Vsatis fies Hypothesis(I1).By(2.7),foru∈Xλ,we have

Step2We will sho wthatJλ,Vsatisfies Hypothesis(I3).Letis finite dimensional subspace ofXλ.Because all the norm in the finite dimensional space are equivalent,then for anyu∈,one can get

for some positive constantC0>0.Also,sp0 such thatJλ,V(u)≤0 onX?λBR.

Step 3We will prove that there exists a sequence{Υn}?(0,+∞)with Υn≤Υn+1,such that

To this end,in view of the de finition ofcn,one can get

so by the de finiti on of Γn,we get Υn<+∞and Υn≤Υn+1.

Step 4We will prove that problem(1.1)has at leastkpair of weak solutions.To this e nd,using the similar argument in the proof of Theorem 1 in[31],fork≥1,by choosing.So,by Step 3,one can get

Hence,by Theorem 2.6 and Proposition 9.30 in[28],the levelc1≤c2≤···≤ckare critical values ofJλ,V.Ifcj=cj+1,then by Theorem 2.6 and Remark 2.12 in[1],Kcjcontains in finitely many distinct points and therefore problem(1.1)has in finitely many weak solutions.Therefore,problem(1.1)has at leastkpair of weak solutions.