# Multiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractionalp-Laplacian of Order s

Authors: M. Khiddi
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractionalp-Laplacian of order s. We show the above result when λ>0 is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.


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## Body

## 1. Introduction

This work is concerned with the existence of weak solutions of the following critical fractionalp-Laplacian problem:(1)-Δpsu=ups∗-2u+λur-2uinΩ,u=0on∂Ω,where Ω is a smoothly bounded domain of RN, N≥sp, 0<s<1, 1<r<p<ps∗≔Np/N-sp is the fractional critical Sobolev exponent, and λ is positive parameter.( - Δ ) p s denotes the fractional p-Laplacian operator defined on smooth functions by (2)-Δpsux=2limε↘0⁡∫RN∖Bεxux-uyp-2ux-uyx-yN+spdy,x∈RN.This definition is consistent, up to a normalization constant depending on N and s, with the usual definition of the linear fractional Laplacian operator (-Δ)s when p=2. Let us recall the weak formulation of problem (1). Let us set (3)
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pby the Gagliardo seminorm of the measurable function u:RN→R, and let (4)Ws,pRN=u∈LpRN:up,s<∞be the fractional Sobolev space endowed with the norm (5)us,p=upp+up,sp1/p,where |·|p is the norm in Lp(RN). We work in the closed linear subspace (6)XpsΩ=u∈Ws,pRN:u=0a.e.inRN∖Ω,equivalently renormed by setting ‖·‖s,p=[·]s,p, which is uniformly convex Banach space. We note that the embedding (7)XpsΩ↪LrΩis continuous for r∈[1,ps∗] and compact for r∈[1,ps∗), where ps∗=pN/N-ps is the fractional critical Sobolev exponent (note that when s=1 the above exponent reduces to the classical Sobolev exponent 2∗). As in the classical case the technical problem posed by such an exponent is that the Sobolev embedding is not compact. Denote by Ss,p the best Sobolev constant of the immersion Xps(Ω)↪Lps∗(Ω), that is,(8)Ss,p≔inf⁡us,pp:ups∗=1,u∈XpsΩ.The dual space of Xps(Ω) is Xps(Ω)∗.Recently, a great attention has been focused on the study of the fractional Laplacian and nonlocal operators of elliptic; this type arises in both pure mathematical research and concrete applications, such as the thin obstacle problem [1, 2], minimal surfaces [3], phase transitions [4], crystal dislocation [5], Markov processes [6], and fractional quantum mechanics [7]. This is one of the reasons why nonlocal fractional problems are widely studied in the literature in many different contexts (see [8]).Whens=1, our problem becomes a scalar quasilinear elliptic equation as follows:(9)-Δpu=up∗-2u+λur-2uinΩ,u=0on∂Ω.This has been widely studied by many authors. For example, when Ω is bounded, the existence of nontrivial solution was studied (see, e.g., [9, 10]). The typical difficulty in dealing with (9) is that the Sobolev embedding W01,p(Ω)↪Lp∗(Ω) is not compact, so the usual “Convex-Compact” method does not apply directly. As for the existence of infinitely many solutions to (9), by using Lusternik-Schnirelmans theory, J. G. Azorero and I. P. Aloson proved in [11] that if Ω is bounded,1<r<p, and λ>0 is small, then (9) has infinitely many solutions. Also the main result of [11] was extended to the equation driven by the operator -Δpu-Δqu by G. Li and X. Liang in [12]. G. M. Figueiredo in [13] generalized the same result of [11] to the elliptic equation generated by the operator -div⁡(a(|∇u|p)|∇u|p-2∇u). Several works have been devoted to study some existence and multiplicity results for fractional problems involving the p-Laplacian operator of the type (1), generalizing therefore some classical results obtained in the scalar case. The reader can find a lot of papers in the literature involving this subject; we cite [14–17]. Our goal is to generalize the results of Garcia Azorero and Peral in [11] to the case of the fractional p-Laplacian (-Δ)ps on a bounded domain.Two major difficulties arise which have to be dealt with in order to reach the desirable conclusions.First off, it is hard to prove the existence of infinitely many negative energy solutions for our equation by using the variational method becauseFλ does not satisfy the (PS) conditions, more precisely, because the problem in question incorporates critical exponents.Secondly, the functionalFλ is not bounded from below, so in order to comfortably follow through with our plan, we have to introduce an appropriate truncation to the problem, the choice of which is of utmost importance to the results we get in this paper.Theorem 1.
Assume that1<r<p<ps∗. Then there exists λ0∈R+∗ such that, for each λ∈(0,λ0), problem (1) has infinitely many solutions with negative energy.Theorem1 is new as far as we know and it generalizes a similar result in [11] for the fractional p-Laplacian (-Δ)ps type problem. We mainly follow the way in [11] to prove our main result. The paper is organized as follows. In Section 2, we show that the (PS)c conditions hold for the related energy functional in certain critical levels. That is, we give in a precise range of compactness for the energy functional related. In Section 3, under the assumptions of Theorem 1 and by application of Ljusternik-Schnirelmann methods, we establish the existence of infinitely many solutions with λ>0 small enough.


## 2. The(PS)c Condition for the Associated Functional

We recall that a weak solution for problem (1) is a function u:Ω→R, u∈Xps(Ω) such that (10)∫R2Nux-uyp-2ux-uyφx-φyx-yN+spdxdy-λ∫Ωur-2uφdx-∫Ωups∗-2uφdx=0,∀φ∈XpsΩ.Now let us consider the functional Fλ:Xps(Ω)→R defined by(11)Fλu=1pus,pp-λrurr-1ps∗ups∗ps∗.Note that the functional Fλ∈C1(Xps(Ω),R) and its derivative at u∈Xps(Ω) are given by (12)Fλ′u,φ=∫R2Nux-uyp-2ux-uyφx-φyx-yN+spdxdy-λ∫Ωur-2uφdx-∫Ωups∗-2uφdx,for every φ∈Xps(Ω). Thus, the weak solutions of problem (1) are precisely the critical points of the energy functional Fλ. Since problem (1) has a variational structure, the proof of the main result (Theorem 1 and its consequences) reduces to finding critical points of the functional by using suitable abstract approaches. As usual in the critical case, the difficulty related to the variational formulation of (1) is the lack of compactness of the injection of the fractional Sobolev space Xps(Ω) in Lps∗(Ω). To overcome this difficulty in treating problem (1), we show that even if the functional Fλ does not verify globally the Palais-Smale condition, it satisfies such a condition in a suitable range related to the best fractional critical Sobolev constant Ss,p noted in (8).The Nehari manifold associated withFλ is given by (13)Nλ≔u∈XpsΩ∖0:Fλ′u,u=0,where Fλ′ denotes the Gâteaux derivative of Fλ.Definition 2.
(i)
Forc∈R, a sequence {un}⊂Xps(Ω) is a (PS)c for Fλ if Fλ(un)=c+o(1) and Fλ′(un)=o(1) strongly in Xps(Ω)∗ as n→+∞, where Xps(Ω)∗ is the dual of Xps(Ω). (ii)
F λ satisfies the (PS)c condition in Xps(Ω) if any (PS)c sequence for Fλ contains a convergent subsequence.The first step for the(PS)c sequence to hold is bounded.Lemma 3.
Letc∈R. If {un} is (PS)c- sequence for Fλ, then {un} is bounded in Xps(Ω).Proposition 4.
There exists aK>0 such that, for any λ>0 and (14)c≤sNSs,pN/sp-Kλps∗/ps∗-r,the functional Fλ satisfies (PS)c condition.Proof.
Let{un} be a sequence in Xps(Ω) such that(15)Fλun=1puns,pp-λrunrr-1ps∗unps∗ps∗=c+o1,(16)Fλ′un,v=∫R2Nunx-unyp-2unx-unyvx-vyx-yN+spdxdy-λ∫Ωunr-2unvdx-∫Ωunps∗-2unvdx=ouns,p,as n→∞, for all v∈Xps(Ω). Then(17)sNunps∗ps∗-1λ1r-1punrr=Fλun-1pFλ′un,un=ouns,p+O1.From (17) and the Hölder inequality, it is implied that (un) is bounded in Xps(Ω). Up to a subsequence, this implies the following:un⇀uinXps(Ω),unx→uxa.einΩ,unx-unyp-2unx-uny/x-yN+sp/p′isboundedinLp′(R2N),un→uinLr(Ω),1≤r<ps∗,unx-unyp-2unx-uny/x-yN+sp/p′isboundedinLp′R2Nandconvergestoux-uyp-2(u(x)-u(y))/x-yN+sp/p′,a.einR2N.
So(18)∫R2Nunx-unyp-2unx-unyvx-vyx-yN+spdxdy→∫R2Nux-uyp-2ux-uyvx-vyx-yN+spdxdyas n→∞.
Moreover(19)∫Ωunr-2unvdx→∫Ωur-2uvdxand∫Ωunps∗-2unvdx→∫Ωups∗-2uvdxSo passing to the limit in (16) shows that u∈Xps(Ω) is a weak solution of (1). Setting vn=un-u, we have(20)vns,pp=uns,pp-us,pp+o1By Brezis-Lieb’s Lemma [18], we get(21)vnps∗ps∗=unps∗ps∗-ups∗ps∗+o1.(22)c+o1=1pvns,pp+1pus,pp-1ps∗vnps∗ps∗-1ps∗ups∗ps∗-λrurr,as n→+∞. Taking v=un in (16) and Brezis-Liebs Lemma again, we have(23)vns,pp=λurr+vnps∗ps∗+ups∗ps∗+o1Since un is bounded in Xps(Ω) and converges to u in Lp(Ω), testing (16) with v=u gives(24)us,pp=λurr+ups∗ps∗.It follows from (23) and (24) that (25)vns,pp=vnps∗ps∗+o1We suppose that (26)limn→∞vns,pp=l=limn→∞vnps∗ps∗By the definition of the best constant Ss,p given in (8), we have (27)vns,pp≥Ss,pvnps∗p,so(28)l≥Ss,plp/ps∗.If l=0, then the lemma is proved. If l>0, then (28) implies that(29)l≥Ss,pN/sp.From (29) and (22), we have (30)c=1pl+1pus,pp-1ps∗l-1ps∗ups∗ps∗-λrurr=sNl+1p-1ps∗ups∗ps∗+λ1p-1rurr≥sNSs,pN/sp+sNups∗ps∗+λ1p-1rurr≥sNSs,pN/sp+sNups∗ps∗+λ1p-1rΩps∗-r/ps∗ups∗r.
We consider the following functionf(x)=s/Nxps∗-λ(1/r-1/p)|Ω|ps∗-r/ps∗xr. This function obtains this absolute minimum for x>0 at point x0=λ((p-r)/ps∗-p)|Ω|ps∗-r/ps∗1/ps∗-r, that is,(31)fx≥fx0=-λps∗/ps∗-rΩp-rps∗-pr/ps∗-rp-rpps∗-rrps∗,and let the constant K=|Ω|((p-r)/ps∗-p)r/ps∗-r(p-r/p)(ps∗-r/rps∗) be strictly positive because 1<r<p<ps∗.(32)c≥sNSs,pN/sp+sNups∗ps∗+λ1p-1rΩps∗-r/ps∗ups∗r,≥sNSs,pN/sp-Kλps∗/ps∗-r.This leads to a contradiction with (9). Therefore l=0 and the proof is complete.


## 3. Proof of the Main Result

Under the hypothesis1<r<p<N, using Sobolev’s inequality we obtain(33)Fλu≥hus,pwhere (34)hx=1pxp-1ps∗Ss,pps∗/pxps∗-λrCp,rxrand where Cr,p is a positive constant independent of u∈Xps(Ω). An easy computation shows that, for all 0<λ<λ0=Cp,r-1ps∗-p/ps∗-r(p-r/ps∗-r)Ss,pps∗/pp-r/ps∗-p, the real valued function x↦h(x) has exactly two positive zeros denoted by R0 and R1, and the point R is where h attains its nonnegative maximum and verifies R0<R<R1.We now introduce the following truncation of the functionalFλ. Take the nonincreasing function τ:R+→[0,1] and C∞(R+) such that(35)τx=1ifx≤R0,τx=0ifx≥R1.Let φ(u)=τ(‖u‖s,p). We consider the truncated functional(36)F~λu=1pus,pp-λr∫Ωurdx-1ps∗∫Ωups∗φudx.Similar to (33), we have(37)F~λu≥h¯us,pwhere(38)h¯x=1pxp-1ps∗Ss,pps∗/pxps∗τx-λrCp,rxr Clearly,(39)h¯x≥hxfor x≥0 and h¯(x)=h(x) if 0≤x≤R0, h¯(x)≥0, if R0<x≤R1 and if x>R1, h¯(x)=xr(1/pxp-r-λ/rCp,r) is strictly increasing and so h¯(x)>0, if x>R1. Consequently(40)h¯x≥0forx≥R0.We have the following result.Lemma 5.
This lemma can be expressed as three assertions:(1)
F~λ∈C1(Xps(Ω),R) is even.(2)
IfF~λ(u0)≤0 then ‖u0‖s,p<R0. Moreover, F~λ(u)=Fλ(u) for all u in a small enough neighborhood of u0.(3)
There existsλ0>0, such that if 0<λ<λ0, then F~λ verifies a local Palais-Smale condition for c≤0.Proof.
Sinceφ∈C∞ and φ(u)=1 for u near 0, F~λ∈C1(Xps(Ω),R) and assertion (1) holds.
By takingF~λ(u0)≤0, we can deduce from (37) that(41)h¯u0s,p≤0,and by (40) and (41) we have(42)u0s,p<R0,and (2) holds.
For the proof of (3), let{un}⊂Xps(Ω) be a (PS)c sequence F~λ, with c<0. Then we may assume that F~λ(un)<0, F~λ′(un)→0. By (8) in Lemma 5 there exists λ0>0 such that 0<λ<λ0, ‖un‖s,p<R0, so F~λ(un)=Fλ(un) and F~λ′(un)=Fλ′(un). By Proposition 4, Fλ satisfies (PS)c condition for c<0, so there is a subsequence {un} such that un→u in Xps(Ω). Thus F~λ satisfies (PS)c condition for c<0.We will use the genus of symmetric set inXps(Ω), where the genus γ(A) is the smallest integer m, such that there exists an odd map (43)ϕ∈CA,Rm∖0,where A is a closed symmetric set in X that does not contain zero (see [19]).It is possible to prove the existence of level sets ofI~λ with arbitrarily large genus, more precisely,Lemma 6.
∀ n ∈ N    ∃ ϵ ( n ) > 0 such that(44)γu∈XpsΩ:F~λu≤-ϵn≥n.Proof.
Letn∈N. we consider En to be subspaces of Xps(Ω) with En being an n-dimensional subspace of Xps(Ω). Let {un}∈En with norm ‖un‖s,p=1. For 0<ρ<R0¯(45)F~λρun≤1pρp-ρps∗ps∗∫Ωunps∗dx-λrρr∫Ωunrdxwe define (46)αn≔inf⁡∫Ωunps∗dx:un∈En,uns,p=1>0,and(47)βn≔inf⁡∫Ωunr:un∈En,uns,p=1>0.
By using the definitions ofαn, βn, and inequality (45), we obtain (48)F~λρun≤1pρp-αnps∗ρps∗-λβnrρr.Then, there exists ϵ(n)>0 and 0<ρ<R0 such that (49)F~λρu≤-ϵnfor u∈En and ‖un‖=1.
LetSη={u∈Xps(Ω)/‖u‖=η}, so(50)Sη∩En⊂u∈XpsΩ/F~λu≤-ϵn,therefore, by the properties of the genus (see [19])(51)γu∈XpsΩ/F~λu≤-ϵ≥γSη∩En≥n.We are now in a position to prove our main result.Proof of Theorem1.
Forn∈N, we define(52)Γn=A⊂XpsΩ-0/Aisclose,A=-A,γA≥n.Let us set(53)cn=minA∈Γn⁡maxu∈A⁡I~λu,and(54)Kc=u∈XpsΩ:F~λ′u=0,F~λu=c,and suppose 0<λ<λo, where λ0 is the constant given by Lemma 5.(55)F~λ-ϵ=u∈XpsΩ/F~λu≤-ϵ.By Lemma 6 there exists ϵ(n)>0 such that γ(F~λ-ϵ)≥n, for all n∈N. Because F~λ(u) is continuous and even, F~λ-ϵ∈Γn, then cn≤-ϵ(n)<0 for all n in N. But F~λ is bounded from below; hence cn>-∞ for all n in N.
Let us assume thatc=cn=cn+1=⋯=cn+r. Note that c<0; therefore, F~λ verifies the Plais-Smale condition in c, and it is easy to see that Kc is a compact set.
Ifγ(Kc)≤r, there exists a closed and symmetric set U verifying Kc⊂U, such that γ(U)≤r. By the deformation lemma (see [20]), we have an odd homeomorphism η:X→X, such that η(F~λc+δ-U)⊂F~λc-δ, for some δ>0. By definition, (56)c=cn=infA∈Γn+rsupu∈AF~λu.There exists then A∈Γn+r, such that supu∈AF~λ(u)<c+δ, i.e., A⊂F~λc+δ, (57)ηA-U⊂ηF~λc+δ-U⊂F~λc-δ.But γ(A-U¯)≥γ(A)-γ(U)≥n, and γ(η(A-U¯))≥γ(A-U¯))≥n.
Then,η(A-U¯)∈Γn. And this is a contradiction; in fact, η(A-U¯)∈Γn implies supu∈η(A-U¯)F~λ(u)≥cn=c.
So we have proved thatγ(Kc)≥r+1. We are now ready to show that Fλ has infinitely many critical point solutions. Note that cn is nondecreasing and strictly negative. We distinguish two cases.
Case 1. Suppose that there are 1<n1<⋯ni<⋯, satisfying (58)cn1<⋯<cni<⋯.In this case, we have infinitely many distinct critical points.
Case 2. We assume in this case that, for some positive integer n0, there is r≥1 such that c=cn0=cn0+1=⋯=cn0+r; then γ(Kcn0)≥r+1 which shows that Kcn0 contains infinitely many distinct elements. Since F~λ(u)=Fλ(u) if F~λ(u)<0, we see that there are infinitely many critical points of Fλ(u). The theorem is proved.



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