Mingqi XIANG (向明啟)

College of Science，Civil Aviation University of China，Tianjin 300300，China

E-mail:xiangmingqi_hit@163.com

Vicen?iu D.R?DULESCU

Faculty of Applied Mathematics，AGH University of Science and Technology，al.Mickiewicza 30，30-059 Krak′ow，Poland Department of Mathematics，University of Craiova，Street A.I.Cuza No.13，200585 Craiova，Romania Institute of Mathematics，Physics and Mechanics，Jadranska 19，1000 Ljubljana，Slovenia

E-mail:radulescu@inf.ucv.ro

Binlin ZHANG (張彬林)?

College of Mathematics and Systems Science，Shandong University of Science and Technology，Qingdao 266590，China

E-mail:zhangbinlin2012@163.com

Abstract This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely，we study the following nonlocal problem: whereis the generalized fractional p-Laplacian operator，N≥1，s∈(0，1)，α1，α2，β∈R，Ω?RN is a bounded domain with Lipschitz boundary，and M:，f:Ω→R are continuous functions.Firstly，we introduce a variational framework for the above problem.Then，the existence of least energy solutions is obtained by using variational methods，provided that the nonlinear term f has (θp-1)-sublinear growth at infinity.Moreover，the existence of in finitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally，we obtain the existence and multiplicity of solutions if f has (θp-1)-superlinear growthat infinity.The main features of our paper are that the Kirchhofffunction may vanish at zero and the nonlinearity may be singular.

Key words Fractional Kirchhoffequation;singular problems;variational and topological methods

1 Introduction and Main Results

LetN≥1，p≥1，q≥1，τ＞0，0≤a≤1，α，β，γ∈R be such that

In the casea＞0，we assume in addition that，withγ=aσ+(1-a)β，0≤α-σ≤1 if

Caffarelli，Kohn and Nirenberg[5]proved the following well-known Caffarelli-Kohn-Nirenberg inequality:

In particular，ifa=1，this inequality becomes

After that，existence and multiplicity of solutions for singular elliptic problems have been investigated by using the Caffarelli-Kohn-Nirenberg inequality.Indeed，due to the Caffarelli-Kohn-Nirenberg inequality，one can study the existence and multiplicity of solutions for some singular elliptic equations like

wherea(x) is a nonnegative function satisfyingandbis a function satisfying.For instance，F(xiàn)elli and Schneider in[17]considered the equation

The authors obtained the existence of positive solutions and non-radial solutions asεsmall enough.Ghergu and Rdulescu[16]studied the singular elliptic equation:

Under suitable assumptions onK，the authors obtained two distinct solutions asλsmall enough by using Ekeland’s variational principle and the mountain pass theorem.In[10]，Chu et al.studied the existence and the qualitative properties of solutions for the singularp-Laplacian type problem

where the operator div (|x|-βa(x，?u)) is a general form of the singularp-Laplacian div (|x|αp|?u|p-2?u) andfsatisfies (p-1)-sublinear growth at infinity.The authors obtained two nontrivial solutions by using variational methods.In[9]，Caristi et al.discussed the following nonlocal degenerate problem:

whereM:R+→R+is a continuous functionn satisfying thatm0otα-1≤M(t)≤m1tα-1for allt∈R+，wherem1＞m0and 1＜α＜minand the nonlinear termfsatisfies the following conditions:

(A1) there exists a constantsuch that 0＜νF(t)≤tf(t) for allt∈R{0};

(H1)=0.

Under the above conditions，the authors obtained the existence and multiplicity of solutions.However，it seems that assumptions (A1) and (H1) can not hold simultaneously.The paper[21]extended the Caffarelli-Kohn-Nirenberg inequality to the case of variable exponent Sobolev spaces and obtained the existence of solutions for a class of singularp(x)-Laplacian equations by using variational methods.

The issue of the Caffarelli-Kohn-Nirenberg inequalities in fractional Sobolev spaces is quite delicate.Very recently，Nguyen and Squasssina in[28]proved that the following fractional Caffarelli-Kohn-Nirenberg inequality (see also[1]for a special case):Lets∈(0，1)，α1，α2，α∈R withα1+α2=α，andN≥1，p＞1，q≥1，τ＞0，0＜a≤1，β，γ∈R be such that

In the casea＞0，assume in addition that，withγ=aσ+(1-a)β，0≤α-σandα-σ≤1 ifUnder the above assumptions，Nguyen and Squassina in[28]proved that if，then

Inspired by the above works，in this paper，we study the following singular fractional Kirchhofftype problem:

whereN≥1，s∈(0，1)，α1，α2∈R，Ω?RNis a bounded domain with Lipschitz boundary containing zero，M:[0，∞)→[0，∞) is a continuous function，f:Ω→R is a continuous function，andis the generalized fractionalp-Laplacian operator which，up to a normalization constant，is defined as follows:

for allu，.Especially，asα1=α2=0 andp=2，the above operator reduces to the well-known fractional Laplace operator (-Δ)s.Furthermore，ifs→1-，then (-Δ)sbecomes the classic Laplace operator-Δ(see[14，Proposition 4.4]).

Since the pioneering work of Caffarelli and Silvestre in[7]，a lot of attention has been attracted to investigate problems involving fractional Laplace operator.Especially，much effort has been focused on the subcritical and critical growth of the nonlinearities，which lead us to study various variational problems using the critical point theory.Problems like (1.2) appeared in many fields of real world，for example，continuum mechanics，phase transition phenomena，population dynamics，minimal surfaces and anomalous diffusion.In fact，fractional Laplace operator can be viewed as the typical outcome of stochastically stabilization of Lévy processes;we refer to[2，6，14，20]for more details.

Problem (1.2) also involves the study of Kirchhoff-type problems.In fact，such problems arise in various models of physical and biological systems.In particular，the existence results concerning Kirchhoff-type problems are more and more abundant in recent years.More precisely，Kirchhoffin[19]established a model governed by the equation

whereu=u(x，t) denotes the lateral displacement，Eis the Young modulus，ρis the mass density，his the cross-section area，Lis the length andp0is the initial axial tension.In fact，Equation (1.3) extends the classical D’Alembert wave equation based on a physical consideration;that is，enclosing the effects of the changes in the length of the strings during the vibrations.In particular，F(xiàn)iscella and Valdinoci in[18]proposed a stationary Kirchhoffmodel involving the fractional Laplacian by investigating the nonlocal aspect of the tension;see[18，Appendix A]for further details.

Throughout the paper，without explicit mention，we assume thatM:is a continuous function and verifies (M0) or (M1) and (M2) as below.

(M0) There existm0＞0 andθ＞1 such thatM(t)≥m0tθ-1for allt≥0;

(M1) For anyd＞0 there existsκ:=κ(d)＞0 such thatM(t)≥κfor allt≥d;

(M2) There existsθ∈(1，N/N-sp) such that

A simple example ofMis given byM(t)=a0+b0θtθ-1for allt≥0 and someθ＞1，wherea0，b0≥0 anda0+b0＞0.WhenMis of this type，problem (1.2) is called to be degenerate ifa=0，while it is named non-degenerate ifa＞0.In recent years，Kirchhoff-type fractional problems have triggered more and more attention.Existence results for nondegenerate Kirchhoff-type fractional Laplacian problems were given，for example，in[30，32].While some recent existence results concerning the degenerate case of Kirchhoff-type fractional Laplacian equations were obtained;see[4，8，12，22–25，31，33，34]and references therein.It is worth pointing out that the degenerate case is rather interesting and is treated in some famous works concerning Kirchhofftheory;see for instance[13].From a physical point of view，it seems rational to describe a realistic model byM(0)=0，which means that the base tension of the string vanishes.

Throughout the paper，we assume thatf:Ω→R is a continuous function.In the following，we enumerate the assumptions concerning the nonlinear termf，but keep in mind that they will not be fulfilled simultaneously:

(f0)fis odd，that is，f(-t)=-f(t) for allt∈R;

(f1)

(f2) there existsq∈(1，p) such thatF(t)≥|t|q，where

(f3) there existq＞θpandC＞0 such that

(f4) there existμ＞θpandT＞0 such thatfsatisfies the Ambrosetti-Rabinowtiz type condition，i.e.，

A simple example of functionfsatisfying (f1)-(f2) is given by

where 1:=Np/(N-sp).

Remark 1.1From (f1) one can deduce thatfis (θp-1)-sublinear at infinity，while from (f4) one can deduce thatfis (θp-1)-superlinear at infinity.

Definition 1.2We say thatuis a (weak) solution of problem (1.2)，if it holds that

We always assume thats∈(0，1)，α1，α2∈R，α=α1+α2，N≥1，and Ω?RNis a bounded domain with Lipschitz boundary and 0∈Ω.Now we are in a position to introduce two existence results involving the case that the nonlinearityfis (θp-1)-sublinear at infinity.

Theorem 1.3Assume thatMfulfills (M0) andfsatisfies (f1)–(f2).Ifβ＞(α-s)θp+N(θ-1)，then problem (1.2) has a least energy solution inwith negative energy.

Moreover，we get the existence of in finitely many solutions of problem (1.2).

Theorem 1.4Assume thatMfulfills (M0) andfsatisfies (f0)，(f1) and (f2).Ifβ＞(α-s)θp+N(θ-1)，then problem (1.2) has in finitely many solutions inwith negative energy.

We also obtain the existence and multiplicity of solutions for problem (1.2) when the nonlinearityfis (θp-1)-superlinear at infinity.

Theorem 1.5Assume thatMfulfills (M1)–(M2) andfsatisfies (f3)–(f4). Ifβ＞(α-s)q+N(q/p-1)，then problem (1.2) admits a nontrivial mountain pass solution in

Theorem 1.6Assume thatMfulfills (M1)–(M2) andfsatisfies (f0) and (f3)–(f4).Ifβ＞(α-s)q+N(q/p-1)，then problem (1.2) has in finitely many solutions in

Remark 1.7Ifα1=α2=α，then we can defineas follows:for anyx∈RN

along anyu∈(RN).

To the best of our knowledge，Theorems 1.3–1.6 are the first existence and multiplicity results for singular Kirchhoff-type problems in the fractional setting.

The rest of the paper is organized as follows:in Section 2，we introduce a variational framework of problem (1.2) and give some necessary properties for the functional setting.In Section 3，we obtain the existence of least energy solution for problem (1.2).In Section 4，the existence of in finitely many solutions is obtained by using genus theory.In Section 5，a mountain pass solution and in finitely many solutions for problem (1.2) are obtained by using the mountain pass theorem and the symmetric mountain pass theorem，respectively.

2 Variational Framework and Preliminary Results

We first provide some basic functional setting that will be used in the next sections.Let 1＜p＜∞and defineas the completion of(Ω) with respect to the norm

Using a similar discussion as in[32]，the spaceis a reflexive Banach space.Let 1＜q＜∞andβ∈R.Define the weighted Lebesgue space

The next fractional Caffarelli-Kohn-Nirenberg inequality will be used later，which was obtained in[28].In fact，by takinga=1 in (1.1)，we have

Theorem 2.1Lets∈(0，1)，1＜p＜N/s，α＞-(N-sp)/pandα-s≤γ≤α.Set.Then there existsC(N，α，s)＞0 such that

Using Theorem 2.1，we have the following embedding theorem:

Theorem 2.2Lets∈(0，1)，1＜p＜N/sandα＞-(N-sp)/p.Thenis continuously embedded inLq(Ω，|x|β)，ifandβ≥(α-s)q+N(q/p-1);the embedding is compact ifandβ＞(α-s)q+N(q/p-1).

ProofIf，then by takingγ=αin Theorem 2.1，the embeddingis continuous.If，then we takeα-s＜γ＜αsuch that

Letu∈.Then by the Hlder inequality，we have

Sinceβ＞(α-s)q+N(q/p-1)，we get

Thus，it follows that

It follows from (2.1) and Theorem 2.1 that

which yields that the embeddingis continuous.

Next we show that the embeddingis compact.To this aim，let{un}be a bounded sequence inFor anyR＞0 withBR(0)?Ω is a ball centered at 0 with radiusR.Then{un}is a bounded sequence inBy Theorem 7.1 in[14]，we obtain that there is a convergent subsequence of{un}inLq(ΩBR(0)).By choosing a diagonal sequence，without loss of generality，we assume that{un}converges inLq(ΩBR(0)) for anyR＞0.

Since the embedding is continuous，we obtain that{un}is bounded in

whereC＞0 denotes various constants independent ofn，m.Asβ＞q(α-s)+N(q/p-q)，it follows thatThus，for anyε＞0 there existsR＞0 such that

Then we can choosen0∈N such that

whereCβ=Rβifβ＜0 andCβ=(diam (Ω))βifβ＞0.Therefore，we conclude

This means that{un}is a Cauchy sequence inLq(Ω，|x|β). □

To study solutions of problem (1.2)，we define the associated functionalI:(Ω，|x|αp)→R as follows:

By assumption (f2)，for anyε＞0 there existsTε＞0 such that

Using (2.3)，β＞(α-s)θp+N(θ-1) and Theorem 2.2，one can verify thatIis well defined，of class，R) and

for allu，v∈(Ω，|x|αp).Clearly，the critical points ofIλare exactly the weak solutions of problem (1.2).

3 Proof of Theorem 1.3

In this section，we always assume thatMsatisfies (M0) andfsatisfies (f1) and (f2).

Let us now recall that the functionalIsatisfies the (PS)ccondition in，if any (PS)csequence，namely a sequence such thatI(un)→candI′(un)→0 asn→∞，admits a strongly convergent subsequence in

In order to study the existence of least energy solutions for problem (1.2) in the sublinear case，we will use the following direct method in the calculus of variations:

Theorem 3.1LetXbe a reflexive Banach space with norm ‖·‖X.Assume that the functionalJ:X→R is

(i) coercive onX，that is，J(u)→∞as ‖u‖X→∞;

(ii) weakly lower semi-continuous onX，that is，for anyu∈Xand any sequence{un}?Xsuch thatun?uweakly inX，

ThenJis bounded from below onXand attains its infimum inX.

Lemma 3.2The functionalIis weakly lower semi-continuous on

ProofWe first show that Φ is weakly lower semi-continuous on.To this aim，we define a functionalH:(Ω，|x|αp)→R as

Chooset0-δ＜t1＜t0＜t2＜t0+δ.By the assumption onM，we know that M is a increasing function.It follows that

Next we prove that Ψ is weakly continuous on.By (f2)，there existsC＞0 such that|f(t)|≤C(1+|t|θp-1) for allt∈R.It follows from Theorem 2.2 thatLθp(Ω，|x|β) is compact forβ＞θp(α-s)+N(θ-1).Using a standard argument，one can deduce that Ψ is weakly continuous on

In conclusion，we obtain thatI(u)=Φ(u)-Ψ(u) is a weakly lower semi-continuous functional on(Ω，|x|αp). □

Lemma 3.3The functionalIis coercive and satisfies the (PS)ccondition.

ProofFor anyε＞0，by (M1) and (2.3)，we obtain that for allu∈(Ω，|x|αp) with ‖u‖≥1，

By Theorem 2.2 andβ＞(α-s)θp+N(θ-1)，there existsC＞0 such that

for allu∈with ‖u‖≥1.Now chooseε=m0/(2C)，we obtain

which together withθp＞1 implies thatI(u)→∞as ‖u‖→∞.Thus we have proved thatIis coercive.

Next we show thatIsatisfies the (PS)ccondition.To this aim，we assume that{un}?is (PS)csequence;that is，I(un)→candI′(un)→0 inSinceIis coercive，{un}is bounded inThus，up to a subsequence，we have

Moreover，by

we deduce

It follows that

By (2.2)，we have

which converges to zero by Theorem 2.2.It follows from (3.1) that

which，together with the fact that〈(u)，un-u〉=0，yields that

Then using a similar discussion as in[32，Lemma 3.6]，we can obtain thatun→uin(Ω，|x|αp).Ifthen up to a subsequence we obtain thatun→0 in□

Proof of Theorem 1.3By Theorem 3.1，Lemmas 3.2 and 3.3，the functionalIhas a global minimizeru∈(Ω，|x|αp)，which is a least energy solution of problem (1.2).Now we prove thatuis nontrivial.Choose a nonnegative functionwith ‖v‖=1 and.Then it follows from the definition ofIand (f2) that

thanks top＞q.Thus，we can choose somet＞0 such thatI(tv)＜0.Then by the minimality ofu，we have

which yields thatuis nontrivial. □

4 Proof of Theorem 1.4

In this section we study the existence of in finitely many solutions of problem (1.2).To this end，we mainly use a classical result due to Clark (see[11]).Before stating our result，we first recall some basic notions on Krasnoselskii’s genus and its properties.

Denote byXa real Banach space.Set

Definition 4.1LetA∈Γ andX=Rk.The genusγ(A) ofAis defined by

If there does not exist such a mapping for anyk≥1，we setγ(A)=∞.Note that ifAis a subset which consists of finitely many pairs of points，thenγ(A)=1.Moreover，γ(?)=0.

Now，we list some necessary results of Krasnoselskii’s genus.

Lemma 4.2(1) LetX=Rkand?Ω be the boundary of an open，symmetric and bounded subset Ω?Rkwith 0∈Ω.Thenγ(?Ω)=k.In particular，let Sk-1be ak-1-dimensional sphere in Rk，thenγ(Sk-1)=k.

(2) LetA?X，Ω be a bounded neighborhood of 0 in Rk，and assume that there exists an odd mappingh∈C(A，?Ω) withha homeomorphism.Thenγ(A)=k.

Theorem 4.3(Clark’s theorem[11]) LetJ∈C1(X，R) be a functional satisfying the (PS)ccondition.Furthermore，let us suppose that

(i)Jis even，i.e.，J(-u)=J(u) for allu∈X，andJis bounded from below;

(ii) there is a compact setA?Γ such thatγ(A)=kand

ThenJpossesses at leastkpairs of distinct critical points，and their corresponding critical values are less thanJ(0).

Proof of Theorem 1.4Set

then it follows from Lemma 3.3 that

SinceAkis finite dimensional，all norms on it are equivalent.Thus there exists a positive constantC＞0 such that

By (f2)，we get

for allu∈Skand 0＜t≤1 small enough，whereSk={u∈Ak:‖u‖=1}.Thus，we can findt*=t(k)∈(0，1) andε*=ε*(k)＞0 such thatI(t*u)≤-ε*＜0 for allu∈Sk.Set.Clearly，is homeomorphic to Sk-1.Thenand so

Sincefis odd，the functionalIis even.In view of Lemma 3.3，we know that all assumptions of Theorem 4.3 are satisfied.Then the functionalIadmits at leastkpairs of distinct critical points.Due to the arbitrary ofk，we obtain the existence of in finitely many critical points ofI.Thus，the proof is complete. □

5 Proofs of Theorems 1.5–1.6

In this section we consider the superlinear case of problem (1.2).Without special mentioning，we always assume thatMsatisfies (M1)–(M2)，andfsatisfies (f3)–(f4).

In the sequel，we shall make use of the following general mountain pass theorem (see[3]):

Theorem 5.1LetXbe a real Banach space andJ∈C1(X，R) withJ(0)=0.Suppose that

(i) there existρ，r＞0 such thatJ(u)≥ρfor allu∈X，with ‖u‖X=r;

(ii) there existse∈Xsatisfying ‖e‖X＞ρsuch thatJ(e)＜0.

Define H={h∈C1([0，1];X):h(0)=1，h(1)=e}.Then

and there exists a (PS)csequence{un}?X.

Now we check that the functionalIsatisfies the mountain geometry properties (i) and (ii).

Lemma 5.2There existr，ρ＞0 such thatI(u)≥ρif ‖u‖=r.

ProofBy (M2)，one can deduce

By (5.1) and (f3)，we obtain

for allu∈(Ω，|x|αp) with ‖u‖≤1.Here we have used the fact that the embedding from(Ω，|x|αp) toLq(Ω，|x|β) is continuous by Theorem 2.2，sinceq∈() andβ＞(α-s)θp+N(θ-1).Sinceq＞θp，we can chooser∈(0，1) small enough such thatThen it follows from (5.2) thatI(u)≥for allu∈(Ω，|x|αp)，with ‖u‖=r. □

Lemma 5.3There existse∈(Ω，|x|αp) with ‖e‖＞rsuch thatI(e)＜0，whereris given by Lemma 5.2.

ProofBy (M2)，we have

Choose a nonnegative functionφ∈(Ω) such that ‖φ‖=1.Then by (f4) and (5.3)，for allτ，withτ＞1，we have

Sinceq＞θp，fixingτ＞0 even large so that we have thatI(e)＜0，wheree=τφ.□

Lemma 5.4The functionalIsatisfies the (PS)ccondition.

ProofLet{un}?(Ω，|x|αp) be such that

asn→∞.We first show that{un}is bounded.Arguing by contradiction，we assume that up to a subsequence，

Using (f4) and (M2)，we deduce

Dividing the above inequality by ‖un‖pand lettingngo to infinity，we obtain

which together withμ＞θpyields a contradiction.Thus，{un}is bounded inWs，p0(Ω，|x|αp).

Then there exist a subsequence of{un}，still denoted by{un}，andusuch that

We first show that

Indeed，by (f3) and the Hlder inequality，we have Using Theorem 2.2，we obtainThen it follows from (5.5) that (5.4) holds true.

Due to the fact that{un}is a (PS)csequence，we have

Then by using a similar discussion as in Lemma 3.3，we conclude that ‖un-u‖→0 asn→∞.In conclusion，the proof is complete. □

Proof of Theorem 1.5By Lemmas 5.2–5.3 and Theorem 5.1，there exists a (PS)csequence{un}such thatI(un)→c，I′(un)→0，whereand H={h∈C1([0，1];(Ω，|x|αp)):h(0)=1，h(1)=e}.Furthermore，by Lemma 5.4，there exist a subsequence of{un}(still denoted by{un}) andu∈(Ω，|x|αp) such thatun→u.Moreover，uis a nonnegative solution of problem (1.2). □

We shall use the following symmetric mountain pass theorem to get the existence of in finitely many solutions of problem (1.2) in the superlinear case:

Theorem 5.5LetXbe a real in finite dimensional Banach space andJ∈C1(X，R) a functional satisfying the (PS)ccondition.Assume thatJsatisfies the following:

(1)J(0)=0 and there existρ，r＞0 such thatJ(u)≥ρfor all ‖u‖X=r;

(2)Jis even;

(3) for all finite dimensional subspace?X，there existsR=R()＞0 such thatJ(u)＜0 for allu∈BR().

ThenJpossesses an unbounded sequence of critical values characterized by a minimax argument.

Proof of Theorem 1.6By (f4)，we have

LetEbe a fixed finite dimensional subspace of(Ω，|x|αp).For anyu∈Ewith ‖u‖=1，and for allt≥1 we have by (5.3) and (5.6) that

asR→∞.Hence there existsR0＞0 so large such thatI(u)＜0 for allu∈E，with ‖u‖=RandR＞R0.Clearly，I(0)=0 andIis even.In view of Lemma 5.2，we know that all assumptions of Theorem 5.5 are satisfied.Thus，problem (1.2) admits an unbounded sequence of solutions. □

AcknowledgementsThe third author of this paper would like to thank Professor Giovanni Molica Bisci for helpful discussions during the preparation of manuscript.