Multiple Solutions for a Fractional p-Laplacian Equation with Concave Nonlinearities

PEI Ruichang

School of Mathematics and Statistics,Tianshui Normal University,Tianshui 741000,China.

Abstract.We investigate a fractional p-Laplacian equation with right-hand-side nonlinearity which exhibits(p-1)-sublinear term of the form λ|u|q-2,q＜p(concave term),and a continuous term f(x,u)which is respectively(p-1)-superlinear or asymptotically(p-1)-linear at infinity.Some existence results for multiple nontrivial solutions are established by using variational methods combined with the Morse theory.

Key Words:Fractional p-Laplacian problems;Morse theory;concave nonlinearities;existence and multiplicity of solutions.

1 Introduction

Forp∈(1,∞),s∈(0,1)and smooth functionsu,define

This definition is consistent,up to a normalization constant depending onNands,with the usual definition of the linear fractional Laplacian(-Δ)swhenp=2.In recent years,many authors have focused on the study of these non-local operators of elliptic type,not only the pure mathematical research but also concrete real-world applications,see[1–5]and the references therein.For an elementary introduction of the properties of the fractional Sobolev space,we refer the interested reader to[6].There is currently a rapidly growing literature on problems involving these nonlocal operators. In particular,the eigenvalue problems associated withwas studied in[7,8],regularity theory in[9,10],and existence theory in the subcritical polynomial case in[11–15].

Following the direction,in this paper we consider the problem

where Ω be bounded domain in RNwith smooth boundary,1＜q＜p,λis a real parameter andHere,we briefly introduce the variation formulation of problem(1.1). Let Ω ?RNbe stated as above,and for all 1 ≤r≤∞denote by|·|rthe norm ofLr(Ω).Moreover,let 0＜s＜1＜p＜∞be real numbers,and the fractional critical exponent be defined asifsp＜Nandp*=∞ifsp≥N.The Gagliardo seminorm is defined for all measurable functionu:RN→R by

Define the fractional Sobolev space

endowed with the norm

Our problem is set in the closed linear subspace

which can be equivalently renormed by setting‖·‖=[·]s,p(see Theorem 7.1 in[6]).We know that(X,‖·‖)is a uniformly convex Banach space and that the embedding(Ω)is continuous for all 1 ≤r≤p*and compact for all 1 ≤r＜p*(see Theorem 6.5,7.1 in[6]).The dual space of(X,‖·‖)is denoted by(X*,‖·‖*).The(p-1)-homogeneous nonlinear operatorA:X→X*is defined for allu,φ∈Xby

Thus,a weak solution of problem(1.1)is a functionu∈Xsuch that

for allφ∈X. SinceXis uniformly convex,by Proposition 1.3 in[16],Asatisfies the following compactness condition:If(un)is a sequence inXsuch thatinXand〈A(un),un-u〉→0,thenun→uinX.

Now,we introduce the minimal hypothesis on thef(x,u)of problem(1.1):withf(x,0)=0 and satisfies the growth condition:

for someC＞0 andγ∈(p,p*),whereifps＜Nandp*=+∞ifN≤ps.From condition(H1),it is easy to know that(1.2)is the Euler-Lagrange equation of the functional

where

It is well known that problem(1.1)is the non-local counterpart of quasilinear elliptic partial differential equations of the type

where Δpu=div(|?u|p-2)?uis thep-Laplacian,1＜q＜pandλis a real parameter.

Forp=2 andλ＜0,the existence of nontrivial solutions of(1.3)has been studied by many authors. For example,de Paiva and Massa[17]use the Ambrosetti-Rabinowitz condition(see[18])and assume that

and have proved that there existsλ*＜0 such that(1.3)has at least three nontrivial solutions forλ∈(λ*,0),where···are the eigenvalues of-Δ inIn the case off(x,u)is sublinear or asymptotically linear at infinity,we refer to[17],[19]for details and further references.

Forp＞1,λ＞0,Ambrosetti et al.[20]assume that the nonlinearityfsatisfies the following version of the Ambrosetti-Rabinowitz condition:

(f1)there exist constantsθ＞pandM＞0 such that

and have obtained that problem(1.3)has multiple nontrivial solutions. Whenp=2,f(x,u)is asymptotically linear at infinity,we also refer to[21].

Particular,forp＞1,λ＜0,Sun[22]have obtained three nontrivial solutions of problem(1.3)where the nonlinearityf(x,u)satisfies standard subcritical growth condition,(f1)and the following condition

(f2)there existsμ＜μ1such that

whereμ1is the first eigenvalue of-Δpin the space

Motivated by their work,the purpose of this paper is to study the existence of multiple nontrivial solutions of problem(1.1)where the nonlinearityf(x,u)is(p-1)-superlinear or asymptotically(p-1)-linear at infinity,respectively.Our approach combines minimax methods based on critical point theory with Morse theoretic arguments and truncation techniques.

Here,we impose some assumptions on the nonlinearityf(x,u)besides(H1)as follows.

(H2)uniformly forx∈Ω,wherep(x)≥0 andp∈L∞(Ω)satisfiesp(x)≤λ1for allx∈Ω andp(x)＜λ1on some Ω0?Ω1with|Ω0|＞0,where Ω1:={x∈andλ1＞0 that has an associated eigenfunctionφ1is the first eigenvalue ofwith homogeneous Dirichlet boundary data;

(H3)f(x,t)is(p-1)-superlinear or asymptotically(p-1)-linear at infinity,i.e.

uniformly for allx∈Ω,wherel∈(λ1,+∞];

(H4) There existθ≥1 andC*＞0 such that

where F(x,t)=f(x,t)t-pF(x,t).

Now,we present our main results:

Theorem 1.1.Assume that conditions(H1)-(H4)hold.If l=+∞,then(1.1)has at least three nontrivial solutions for any λ≤0.

Remark 1.1.As far as the problem(1.3)is concerned,this similar result first appears in[22].Compared with the version of Ambrosetti-Rabinowitz condition(f1)used in[22],our conditions(H3)and(H4)are very general.More detailed information for the origin and changing of the generalized(p-1)-superlinear conditions(H2),(H3)will be seen in[23]. In the case ofp=2 andλ=0 for problem(1.1),under conditions(H1)-(H4),two nontrivial solutions can be obtained similarly to[13],but the existence of the third solution has some difficulty.However,using the method in[24],for anyp＞1 andλ＜0,we can provide some information for the critical group of the mountain pass solutions and find the third nontrivial solution.Therefore,Theorem 1.1 improves the results given in[12,13,15].

Theorem 1.2.Assume that conditions(H1)-(H3)hold.If p≥2,by means of the cohomological index,see[11])for some k≥2,then(1.1)has at least three nontrivial solutions for any λ≤0.

Remark 1.2.It is obvious that we deal with the asymptotically(p-1)-linear case of problem(1.1)owing to the condition(H3)withl∈(λk,λk+1).In fact,whenλ=0,this case has been studied in[11],but one will find that they only obtained one nontrivial solution.So our result improves Theorem 6.1 in[11].

The paper is organized as follows.In Section 2,we present some necessary preliminary knowledge about compactness,mountain pass theorem and Morse theory.In Section 3,we prove some lemmas in order to prove our main results.In Section 4,we give the proofs for our main results.In the sequel,the lettercnot having been explained will be used to denote various constant whose exact value is irrelevant.

2 Preliminaries

In this section,we give some preliminary results which will be used in the sequel.First,we recall some definitions for compactness condition and a version of mountain pass theorem.

Definition 2.1.Let(X,‖·‖X)be a real Banach space with its dual space(X*,‖·‖X*)andJ ∈C1(X,R).For c∈R,we say thatJsatisfies the(PS)c condition if for any sequence{un}?E with

there is a subsequence{unk}such that{unk}converges strongly in X.Also,we say thatJsatisfy the(C)c condition if for any sequence{un}?X with

there is subsequence{unk}such that{unk}converges strongly in X.

We have the following version of the mountain pass theorem(see[18,25]):

Proposition 2.1.Let X be a real Banach space and suppose thatJ ∈C1(X,R)satisfies the condition

for some α＜β,ρ＞0and u1∈X with‖u1‖＞ρ.Let c≥β be characterized by

whereΓ={γ∈C([0,1],X),γ(0)=0,γ(1)=u1}is the set of continuous paths joining0and u1.Then,there exists a sequence{un}?X such that

Next,we recall some concepts and results of Morse theory.For the details,we refer to[26].LetXbe a real Banach space and J ∈C1(X,R).K={u∈X|J′(u)=0}is the critical set of J.Letu∈Kbe an isolated critical point of J with J(u)=c∈R,andUbe an isolated neighbourhood ofu,i.e.K∩U={u}.The group

is called the*-th critical group of J atu,where Jc={u∈X|J(u)≤c}.

H*(·,·)is the singular relative homology group of J at infinity is defined by

We denote

Letα＜βbe the regular values of J and set

IfK={u1,u2,...,uk},then there is a polynomialQ(t)with nonnegative integer as its coefficients such that

3 Some Lemmas

Let

where

Then the critical points of J±are exactly the weak solutions of(1.1).

Lemma 3.1.Assume that conditions(H1)and(H2)hold.Then u=0is a local minimum ofJandJ±for any λ≤0.

Proof.First,we claim that if condition(H2)holds,then there exists a constantδ∈(0,1)such that

for allu∈X.

By contradiction,there exists a sequence{un}?Xsuch that

Setit follows that

From condition(H2),we get

Combining above two formulas,we have

Since{vn}is bounded inX,up to a subsequence,the Sobolev embedding theorem(see introduction)give that

From above formulas,we obtain

Thus,since the weakly lower semicontinuity of the norm with the variational characterization ofλ1,we get

Then it follows that

This shows thatv=φ1,mean while,we also obtain

Thus,it happens a contradiction with(H2).

Now,from conditions(H1)and(H2),for given∈＞0 small enough,there existsA1＞0 such that

which implies that

where(Kis the constant of the embeddingthen usingγ＞p,there existsρ＞0 small such that J(u)≥0 as‖u‖≤ρ.Sou=0 is a local minimum of J.The case of J±is similar.

Lemma 3.2.Under conditions(H1),(H2),(H3)with l=+∞and(H4),the functionalsJandJ±satisfy the(C)condition for any λ≤0.

Proof.We only give the proof of J+,the cases of J and J-are similar.Let{un}?Xbe a sequence such that

In order to prove the lemma,we divide two steps to prove it.

Step 1.We first prove that{un}is bounded inX.LetFrom(3.1),we get

where∈n→0 asn→∞,then the boundedness ofcan be directly obtained.For the case of,by contradiction,we assume that→∞asn→∞.Letthen‖vn‖=1.By previous knowledge,up to a subsequence,we have

Case 1.Suppose that,then the Lebesgue measure ofis positive.Using(3.1),we get

which implies that

By(H3),there is a constantM＞0 such that

then we have

On the other hand,forx∈Ω0,asn→∞.Then by the Fatou’s lemma and(H3)we imply that

This combining with(3.7)gives that

This is a contradiction with(3.6).Then this case is impossible.

Case 2.Assume thatv=0,let{tn}?R such that

For anym＞0,we assume that

Thenwn→0 inLq(RN).So from conditions(H1)and(H2),for every∈＞0,we can find a constantC(∈)＞0 such that

which implies that

which deduces that

According to J+(0)=0 andwe havetn∈(0,1),then

Then,from(H4)it follows that

This contradicts the fact thatHence{un}is bounded,that is,there exists a positive constantMsuch that

Step 2.We prove{un}has a convergent subsequence.In fact,we can suppose that

Now,since Ω is a bounded set,again by conditions(H1)and(H2),we can find a constantA2＞0 such that

then

Similarly,sinceSubsequently,we can conclude that

By(3.12),we have

So from property of operatorA,we have

which means that J+satisfies condition(C).

Lemma 3.3.Assume that λ≤0,conditions(H1)(H2),(H3)with l=+∞and(H4)hold.If u±are the isolated nontrivial critical points ofJ±,then we have

Proof.Set

then Jssatisfies the(C)ccondition by Lemma 3.2,andu+is a critical point of Jsfor alls∈[0,1].Now,we claim thatu+is the only critical point of Jsin a neighborhood ofu+for alls∈[0,1].

By contradiction,we suppose that there is a sequencesn∈[0,1]and a critical pointun∈Xof Jssuch that

Sinceu+＞0 is the isolated critical point,we have

Sinceunis the critical point of Jsn,we have

By the previous section of proof of Lemma 3.1,we have

which means that

and it leads to a contradiction.

By the homotopy invariance of the critical groups(see[26]),we have

The case of J-is similar.

Lemma 3.4.Assume that conditions(H1),(H3)with l=+∞and(H4)hold.Then we have

Proof.We only give the proof of J+and the others are similar.

LetandBy(H3),for anyM＞0 there existsc＞0,such thatfor(x,t)∈Ω×R,which implies

for anyu∈S.Using(H4),we have

Choose

Then for anyu∈S,there existst＞1 such that J+(tu)≤a,that is

which together with(3.14),implies

Therefore,by implicit function theorem,there exists a uniqueT∈C(S,R)such that

LetWe construct a strong deformation retractτ:[0,1]×S1→S1which satisfiesif J+(u)≥aandτ(s,u)=uif J+(u)＜a.Hence,It follows from the construction ofτthatis a strong deformation retract ofS1,which is homotopy equivalent to the setS.By the homotopy invariance of homology group,we have

This completes the proof of the lemma.

Lemma 3.5.Under conditions(H2)and(H3)with l∈(λk,λk+1),the functionalsJandJ±satisfy the(PS)condition for any fixed λ.

Proof.We only give the proof of J,the cases of J+and J-are similar.Let{un}?Xbe a sequence such thatSince

for allφ∈X.If|un|pis bounded,we can takeφ=un.By(H2)and(H3),there exists a constantc＞0 such that,a.e.x∈Ω.Sounis bounded inX.If|un|p→+∞,asn→∞,setvn=un/|un|p,then|vn|p=1.Takingφ=vnin(3.15),it follows that‖vn‖is bounded.Without loss of generality,we assumeinX,thenvn→vinLp(Ω).Hence,vn→va.e.in Ω.Dividing both sides of(3.15)by,we get

Then for a.e.x∈Ω,we have

In fact,ifby(H3),we have

Ifv(x)=0,we have

Obviously,hence,lis an eigenvalue ofThis contradicts our assumption.

4 Proof of the main results

In this section,we are ready to give the proof of our main results.

4.1 Proof of the Theorem 1.1.

By Lemma 3.2,we know that J,J±satisfy(C)condition. We know that 0 is a local minimum of J and J±since Lemma 3.1.So,we have

Using the Proposition 2.1 and maximum principle in[11],we get J+(J-)has a critical pointu+＞0(u-＜0),andu±are also the nontrivial critical points of the functional J.Without loss of generality,we assume thatu±are isolated and the only nontrivial critical points of the functional J.

Now we claim that

Indeed,using the methods of[24],we let J+(u+)=c＞0.It follows from the homology exact sequence of the triplewe have

whereA＜0 is a constant.Since 0 is the only critical point of J+in the setby(4.1),we get

Similarly,sinceu+is the only critical point of J+in the setwe have

From Lemma 3.4,we have

From(4.3)to(4.6),we deduce that

The case foru-is similar.

According to the claim and Lemma 3.3,we have

The Morse equality(2.1)witht=-1 implies that

which is a contradiction.Then(1.1)has at least three nontrivial solutions.

4.2 Proof of the Theorem 1.2.

By Lemma 3.1,Lemma 3.5 and the condition(H3)withλk＜l＜λk+1,the functional J+(J-)satisfies mountain pass geometry and compactness condition.Hence,we get two nontrivial solutionsu±.Using similar to the proof of Proposition 7 in[27],we know that

for*∈{0,1}.Thus,similar to the proof of Theorem 1.1,from(4.3),we get

Again using(4.4),(4.5)and above two formulas,we have

The case foru-is similar.Thus,slightly modifying the proof of Lemma 3.3,we obtain

On the other hand,ifλk＜l＜λk+1(k≥2),from[12,Theorem 5.7]and Lemma 3.5,there existsu∈K(J)such thatIt is easy to see thatsince 0 is local minimum of J.Thus,u,u+andu-are three different nontrivial solutions of problem(1.1).

Acknowledgement

This research is supported by the NSFC(Nos.11661070,11764035 and 11571176).