# Existence of Two Positive Solutions for Two Kinds of Fractionalp-Laplacian Equations

Authors: Yong Wu; Said Taarabti
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operatorLKp. Specifically, we get multiple solutions to the following fractional p-Laplacian equations with the help of fibering maps and Nehari manifold. −Δpsux=λuq+ur,u>0inΩ,u=0,inℝN\Ω.. Our results extend the previous results in some respects.


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## Body

## 1. Introduction

In this work, we are concerned with the existence of solutions for a nonlocal integro-differential equation(1)−LKpux=λuq+ur,u>0inΩ,u=0,inℝN\Ω,where Ω is a bounded smooth domain in ℝn,n>ps with s∈0,1,λ>0, the exponents r and q fulfill 0<q<1<r≤ps∗−1 with the critical fractional Sobolev exponent ps∗=np/n−psn>ps, and LKp is a kind of nonlocal integro-differential operator defined by:
(2)LKpux=2limε⟶0+∫ℝN\Bεxux−uyp−2ux−uyKx−ydy,x∈ℝN, and K:ℝN\0⟶0,+∞ is a measurable function with the following property:
(3)γK∈L1ℝNwhereγx=minxp,1,thereexistsak0>0suchthat,Kx≥k0x−N+psforanyx∈ℝN\0,Kx=K−xforanyx∈ℝN\0.In recent years, the existence and multiplicity of solutions of elliptic equations in nonlinear analysis have attracted the attention of many scholars. In particular, problems with regular nolinearities likeuq+λup,p,q>0 and singular nonlinearities u−q+λup,p,q>0. At the same time, elliptic problems can be divided into two categories according to their order: integer order and fractional order.On the one hand, whens=1, in [1], the authors considered a class of semilinear problems with singular nonlinearities. Many results on the existence and multiplicity of solutions for singular problems have appeared in the literature [2–6]. For example, authors have investigated a singular problem with the kind of critical growth in [6],
(4)−Δu=λu−q+u2∗−1,u>0inΩ,u=0on∂Ω,where 0<q<1. They obtained the existence of solutions by means of the Nehari manifold method in a suitable range of λ.On the other hand, in [7], Mukherjee and Sreenadh considered the following critical fractional Laplace operator equations with a singular nonlinearity
(5)−Δsu=λaxu−q+u2s∗−1,u>0inΩ,u=0inℝn\Ω.They showed the existence and multiplicity of positive solutions with respect to the parameter λ for above equation by using variational methods. Furthermore, in [8], they studied a class of critical fractional problems with a lower order perturbation by means of variational and topological methods; precisely, they proved that the number of nontrivial weak solutions is at least twice the multiplicity of the eigenvalue. More details on the critical case of fractional p-Laplace equations can be referred to [9]. In subcritical case, the existence of positive solutions to the following quasi-linear problem
(6)−Δpsu=λgx,u−fx,u,x∈Ω,u=0,x∈ℝN\Ω,is studied by means of truncation and comparison techniques in [10]. Zuo et al. [11] investigated a superlinear fractional elliptic equations; the existence of infinity many solutions is obtained by the fountain theorem in subcritical case. Moreover, they also get at least two solutions for a fractional p-Laplace system by the Nehari manifold method in [12]. We will adopt a new technique, considering both subcritical and critical cases in a more general operator context (see (1)).In order to state our results, let us introduce some notations. The space(7)X=u∣u:ℝN⟶ℝismeasurable,uΩ∈LpΩandux−uyKx−yp∈LpQ,where Q=ℝ2N\CΩ×CΩ with CΩ=ℝN\Ω. The space X is endowed with the norm,
(8)∥u∥X=∥u∥LpΩ+∫Qux−uypKx−ydxdy1/p,and we define the closed linear subspace
(9)X0=u∈X:u=0almosteverywhereinℝN\Ω,with the norm
(10)∥u∥X0=∫Qux−uypKx−ydxdy1/p.Let(11)K:ℝN\0⟶0,+∞,fulfill condition (3). We have that C0∞Ω⊂X0, and X0,∥·∥X0 is a reflexive Banach space (see [13]). Moreover,
(12)X⊂Ws,pΩ,X0⊂Ws,pℝN,where Ws,pΩ is the usual fractional Sobolev space endowed norm
(13)∥u∥Ws,pΩ=∥u∥LpΩ+∫Ω×Ωux−uypx−yn+psdxdy1/p,and the embedding
(14)X0↪Lps∗Ω, is continuous, and there exists a positive constant C0=C0N,p,s such that, for any v∈X0,1<k<ps∗(15)∥v∥LkΩ≤C0∥v∥X0.Definition 1.
We say thatu is a weak solution of problem (1), if u fulfills
(16)∫Qux−uyp−2ux−uyφx−φyKx−ydxdy=∫Ωλu+qx+u+pxφxdx,for all φ∈X0, where u+=maxu,0.The main results of this article are as follows.Theorem 2.
Sets∈0,1,n>ps, K fulfilling condition (3), if 0<q<1,1<r<ps∗, then there exists λ★>0, such that for λ∈0,λ★, equation (1) has at least two positive solutions.Theorem 3.
Sets∈0,1,n>ps, and Ω be an open bounded domain in ℝn with Lipschitz boundary. K fulfilling the condition (3), if 0<q<1,r=ps∗−1, assumes that there exists u0∈X0\0 with u0≥0 almost everywhere in ℝn, such that
(17)supt≥0IK,ps∗tu0<snSKn/ps,where IK,ps∗ will be introduced in Section 2. Then, there exists λ2>0, such that for λ∈0,λ2, problem (1) admits least two solutions.


## 2. Preliminaries

We define the energy functional(18)Jλ:X0⟶ℝ,associated to problem (1) as
(19)Jλu=IK,pu−Iλu,with
(20)IK,pu=1p∫Qux−uypKx−ydxdy−1r+1∫Ωu+r+1xdx,Iλu=λq+1∫Ωu+q+1xdx.We can see thatJλ∈C1X0,ℝ and
(21)Jλ′u,φX0=∫Qux−uyp−2ux−uy×φx−φyKx−ydxdy−λ∫Ωu+qxφxdx−∫Ωu+rxφxdx,for anyφ∈X0.Now, we give the Nehari manifold(22)Nλ=u∈X0\0:Jλ′u,u=0,where , denotes the duality between X0 and its dual space. Thus, u∈Nλ if and only if
(23)uX0p−λ∫Ωu+q+1xdx−∫Ωu+r+1xdx=0.The Nehari manifoldNλ is closely related to the following function φu:t↦Jλtu for t>0 defined by
(24)φut≔Jλtu=tpp∥u∥X0p−λtq+1q+1∫Ωu+q+1xdx−tr+1r+1∫Ωu+r+1xdx.Remark 4.
Setu∈X0\0, then tu∈Nλifandonlyifφu′t=0.
Moreover,(25)φu′t=tp−1∥u∥X0p−λtq∫Ωu+q+1xdx−tr∫Ωu+r+1xdx,(26)φu′′t=p−1tp−2∥u∥X0p−qλtq−1∫Ωu+q+1xdx−rtr−1∫Ωu+r+1xdx.
According to (25) and Remark 4, for u∈Nλ, we have
(27)φu′′1=p−1∥u∥X0p−λq∫u+p+1xdx−r∫Ωu+r+1xdx=p−r−1∫Ωu+r+1xdx+λp−q−1∫Ωu+q+1xdx=p−q−1∥u∥X0p−q+r∫Ωu+r+1xdx=p−r−1∥u∥X0p−λq−r∫Ωu+q+1xdx.
TheNλ is divided into three sets, which are local minimum, local maximum, and local inflection point, respectively, i.e.,
(28)Nλ+=u∈Nλ:φu′′1>0,Nλ+=u∈Nλ:φu′′1>0,Nλ0=u∈Nλ:φu′′1=0.To prove our result, we should start to show the following auxiliary lemmas.Lemma 5.
Ifu0 is a local minimizer of Jλ on Nλ and u0∉Nλ0, then u0 is a critical point of Jλ.
Similar to Theorem 2.3 in [14], we can get this conclusion.
About fibering maps and the Nehari manifold, considering the functionψu:ℝ+⟶ℝ defined by
(29)ψut=t1−q∥u∥X0p−tr−q∫Ωu+r+1dx.
Obviously, for anyt>0,tu∈Nλ if and only if
(30)ψut=λ∫Ωu+q+1dx.
Moreover,(31)ψu′t=1−qt−q∥u∥X0p−r−qtr−q−1∫Ωu+r+1dx,
and moreover, we know thattu∈Nλ, then
(32)tqψu′t=φu′′t.
So,tu∈Nλ+orNλ− if and only if ψu′t>0 (or <0). Assume u∈X0 and u+≠0. In view of (29), ψu fulfills the following properties:
(i)
ψu has a unique critical point at t=tmaxu=1−q∥u∥X0p/r−q∫Ωu+r+1dx1/r−1>0(ii)
ψu↑ on 0,tmaxu and ↓ on tmaxu,+∞(33)limt⟶+∞ψut=−∞.
Further, it follows from∫Ωu+q+1dx>0 that (30) has no solutions if λ fulfills
(34)λ∫Ωu+q+1dx>ψutmaxu=1−qr−q1−q/r−1−1−qr−qr−q/r−1∥u∥X0pr−q/r−1∫Ωu+r+1dx1−q/r−1.
According to (25) and (30) if λ fulfills (34), then φu′t>0. It seems φu′t<0 as λ is sufficiently large. Therefore, tu∉Nλ for any t>0. Moreover, if λ fulfills
(35)0<λ∫Ωu+q+1dx<ψutmaxu,then there exist t1 and t2 with t1<tmaxu<t2, such that
(36)ψut1=ψut2=λ∫Ωu+q+1dx,andψu′t1>0,ψu′t2<0,combining (25) and (30), which imply that φu′t1=φu′t2=0. It follows from (32) that φu′′t1>0,φu′′t2<0, which mean that the fibering map φu admits a local minimum t1u∈Nλ+ and a local maximum at t2u∈Nλ−.


## 3. The Subcritical Case:0<q<1<r<ps∗−1

Firstly, we prove the following lemmas.Lemma 6.
There existsλ⋆>0, such that for any λ∈0,λ⋆, we have Nλ0=∅.Proof.
Using the inverse method, ifNλ0≠∅ for any λ>0. Then,
(37)Jλ′u,uX0=0,φu′′1=0.for u∈Nλ0.
Namely,(38)∥u∥X0p=λ∫Ωu+q+1ds+∫Ωu+r+1dx,and∥u∥X02=λq∫Ωu+q+1ds+r∫Ωu+r+1dx.
Thus,(39)1−q∥u∥X0p=r−q∫Ωu+r+1dx,andr−1∥u∥X0p=λr−q∫Ωu+q+1dx.
Using the Hölder inequality and Remark4, there exist two positive constants C1,C2 such that
(40)∥u∥X0p≤C1∥u∥X0r+1and∥u∥X0p≤λC2∥u∥X0q+1.
It yields thatC11/p−r−1≤∥u∥X0≤λC21/p−q−1. If λ is small enough, then it is impossible. Thus, assuming no, the original set is empty.Lemma 7.
Jλ is coercive and bounded from below on Nλ for λ∈0,λ⋆.Proof.
Letu∈Nλ, (19) and (23) we get
(41)Jλu=1p−1r+1∥u∥X0p−λ1q+1−1r+1∫Ωu+q+1dx.
Using Remark4 and Hölder inequality, we get
(42)∫Ωu+q+1dx≤Cn,q,s,θ,Ω∥u∥X0q+1.
Prove complete due to0<q<1<r.
By Lemmas6 and 7, for any λ∈0,λ⋆, we get Nλ=Nλ+∪Nλ−, and so, Jλ is coercive and bounded from below on Nλ+ and Nλ−. Therefore, we define
(43)αλ+=infu∈Nλ+Jλu,αλ−=infu∈Nλ−Jλu.
We have the following result.Proposition 8.
If0<λ<λ⋆, then the functional Jλ has a minimizer u1 in Nλ+ and satisfies
(1)
Jλu1=infu∈Nλ+Jλu<0.(2)
u1 is a solution of problem (1).Proof.
Since the bounded from below ofJλ on Nλ+, there exists a minimizing sequence uk⊂Nλ+, such that
(44)limk⟶∞Jλuk=infu∈Nλ+Jλu.
We know that the sequenceuk is bounded in X0 by Lemma 7. X0,∥·∥X0 is a Hilbert space (see Lemma 7 in [15]); thus, there exists u1∈X0 such that, up to a subsequence,
(45)∫Qukx−ukyp−2ukx−ukyϕx−ϕyKx−ydxdy⟶∫Qu1x−u2yp−2u1x−u1yϕx−ϕyKx−ydxdyfor∀ϕ∈X0,when k⟶∞. Further, by Lemma 8 in [15], we have
(46)uk⟶u1inLrℝn,uk⟶u1a.eℝn,as k⟶∞, and by ([16], TheoremIV − 9), there exists ℓ∈Lrℝn such that
(47)ukx≤ℓxa.einℝn,for any 1≤r<ps∗=np/n−psn>ps. It follows from the dominated convergence theorem that
(48)∫Ωuk+q+1dx⟶∫Ωu1+q+1dx,and∫Ωuk+r+1dx⟶∫Ωu1+r+1dx,
ask⟶∞. So, there exists t1 such that t1u1∈Nλ+ and Jλt1u1<0. Therefore, we get infu∈Nλ+Jλu<0.
In order to prove thatuk⟶u1 strongly in X0. Still use the arc method if not, then u1X0<liminfk⟶∞ukX0. Hence, for uk∈Nλ+, we get
(49)limk⟶∞φuk′t1=limk⟶∞t1ukX0p−λt1q∫Ωuk+q+1dx−t1r∫Ωuk+r+1dx>t1u1X0p−λt1q∫Ωu1+q+1dx−t1r∫Ωu1x+r+1dx=φu1′t1=0.
That is,φuk′t1>0 for k large enough. Since uk=1,uk∈Nλ+, we infer that φuk′t<0 for t∈0,1 and φuk′1=0 for all k. So, must be t1>1. In addition because φu1t is decreasing on 0,t1, and so,
(50)Jλt1u1≤Jλu1<limk⟶∞Jλuk=infu∈Nλ+Jλu.
Obviously, the above equation is a contradiction. Therefore,uk⟶u1 strongly in X0. It means that
(51)Jλuk⟶Jλu1=infu∈Nλ+Jλuask⟶∞,
i.e.,u1 is a minimizer if Jλ on Nλ+. By Lemma 5, u1 is a solution to problem (1).Proposition 9.
If0<λ<λ1, then Jλ admits a minimizer u2 in Nλ− and satisfies
(1)
Jλu2=infu∈Nλ−Jλu>0.(2)
u2 is a solution to problem (1).Proof.
Since the bounded from below ofJλ on Nλ−, there exists a minimizing sequence u~k⊂Nλ−, such that
(52)limk⟶∞Jλu~k=infu∈Nλ−Jλu.
Similar to Proposition8, there exists u2∈X0 such that
(53)∫Qu~kx−u~kyp−2u~kx−u~kyϕx−ϕyKx−ydxdy⟶∫Qu2x−u2yp−2u2x−u2yϕx−ϕyKx−ydxdyfor∀ϕ∈X0,
ask⟶∞, and
(54)∫Ωu~k+q+1dx⟶∫Ωu2+q+1dx,and∫Ωu~k+r+1dx⟶∫Ωu2+r+1dx,as k⟶∞. Moreover, from the nature of the fibering maps φut, we infer that there exist t1,t2 with t1<tmaxu<t2 such that t1u∈Nλ+,t2u∈Nλ−, and Jλt1u≤Jλtu≤Jλt2u.
Next, we show thatu~k⟶u2 strongly in X0. If not, then u2X0<liminfk⟶∞u~kX0. Thus, for u~k∈Nλ−, we have Jλu~k≥Jλtu~k for all t≥tmaxu, and
(55)Jλt2u2=t2ppu2X0p−λt2q+1q+1∫Ωu2+q+1dx−t2r+1r+1∫Ωu2+r+1dx<limk⟶∞t2ppu~kX0p−λt2q+1q+1∫Ωu~k+q+1dx−t2r+1r+1∫Ωu~k+r+1dx=limk⟶∞Jλt2u~k≤Jλu~k=infu∈Nλ−Jλu,in a similar way, we still can get a contradiction. Thus, u~k⟶u2 strongly in X0. It means that
(56)Jλu~k⟶Jλu2=infu∈Nλ−Jλuask⟶∞.
Namely,u2 is a minimizer if Jλ on Nλ−.u2 is a solution to problem (1) according to Lemma 5.Proof of Theorem 10.
We obtain that problem (1) has two solutions u1∈Nλ+ and u2∈Nλ− in X0 due to the Propositions 8 and 9 and Lemma 5; moreover, we know that two solutions are distinct since Nλ+∩Nλ−=∅.


## 4. The Critical Case:0<q<1,r=ps∗−1

For the critical case, since the embeddingX0↪Lps∗Ω is not compact, then the energy functional does not satisfy the Palais-Smale condition globally, but it is true for the energy functional in a suitable range related to the best fractional critical Sobolev constant in the embedding X0↪Lps∗ℝn. For this, we define fractional Sobolev best constant SK as
(57)SK=infv∈X0\0∫ℝ2nvx−vypKx−ydxdy∫Ωvxps∗p/ps∗forv∈X0\0.Before we give the Proof of Theorem13, we start by some auxiliary results. Firstly, using the same proofs of Lemma 6, we deduce that there exists λ∗>0 such that Nλ0=∅ for each λ∈0,λ∗. Also, it is clear that Jλ is coercive and bounded from below on Nλ for λ∈0,λ∗ by Lemma 7. So, for any λ∈0,λ∗, we also obtain that Nλ=Nλ+∪Nλ−, and Jλ is coercive and bounded from below on Nλ+ and Nλ−. We define
(58)α~λ+=infu∈Nλ+Jλu,α~λ−=infu∈Nλ−Jλu.Proposition 11.
Letuk⊂X0 be a PSc sequence for Jλ with
(59)c<snSKn/ps−Mλps∗/ps★−q,then there exists a subsequence of uk, which converges strongly in X0, where SK is defined in (57) and M>0 is defined by
(60)M=2n−n−2sq+11−q4q+11−qn−2s4sq+1/ps★−q+1Ω.Proof.
It follows fromuk is bounded in X0 that there exists u∞∈X0 such that uk⟶u∞ weakly in X0, that is
(61)∫Qukx−ukyp−2ukx−ukyϕx−ϕyKx−ydxdy⟶∫Qu∞x−u∞yp−2u∞x−u∞yϕx−ϕyKx−ydxdyfor∀ϕ∈X0,as k⟶∞. Moreover, using the same arguments as lemma 9 ([17]), we get that
(62)uk⟶u∞weaklyinLps∗ℝn;uk⟶u∞inLrℝn;uk⟶u∞a.e.inℝn,as k⟶∞, and by ([16], TheoremIV − 9], there exists ℓ∈Lrℝn such that
(63)ukx≤ℓxa.e.inℝn,for any 1≤r<ps∗=np/n−psn>ps. Then, using dominated convergence theorem, we have that
(64)∫Ωuk+q+1dx⟶∫Ωu∞+q+1dx.
Also, by the same method as in ([18], Lemma 1.32), we get
(65)∫Qukx−ukypKx−ydxdy⟶∫Qukx−u∞x−uky+u∞ypKx−ydxdy+∫Qu∞x−u∞ypKx−ydxdy+o1,∫Ωukx+ps∗dx=∫Ωuk−u∞x+ps∗dx+∫Ωu∞x+ps∗dx+o1,as k⟶∞. Then,
(66)Jλ′uk,ukX0=∫Qukx−ukypKx−ydxdy−λ∫ukx+q+1dx−∫Ωukx+ps∗dx=∫Ωukx−u∞x−uky+u∞ypKx−ydxdy+∫Qu∞x−u∞ypKx−ydxdy−λ∫Ωukx+q+1dx−∫Ωuk−u∞x+ps∗dx+∫Ωu∞+rdx+o1+o1=∫Quk−u∞x−uk−u∞ypKx−ydxdy−∫Ωuk−u∞x+ps∗dx+Jλ′u∞,u∞X0+o1.
ByJλ′u∞,u∞X0=0 and Jλ′uk,ukX0⟶0 as k⟶∞, we know that
(67)uk−u∞X0p=∫Quk−u∞x−uk−u∞ypKx−ydxdy⟶b,∫Ωuk−u∞x+ps∗dx⟶b,ask⟶∞.
Ifb=0, is clearly true. If b>0, in view of the definition of SK in 17, we get
(68)uk−u∞X0p≥SK∫Ωuk−u∞x+ps∗dxp/ps2.
Thus, we haveb≥SKbp/ps∗. That is, b≥SKn/ps. On the other hand, we have
(69)c=limk⟶∞Jλuk=limk⟶∞1pukX0p−λ1q+1∫Ωukx+q+1dx−1r+1∫Ωukx+p★dx≥Jλu∞+snSKn/ps.
By the assumption thatc<s/nSKn/ps, we have Jλu∞<0. In particular, u∞≠0 and
(70)0<1pu∞X0p<1ps∗∫Ωu∞x+ps∗dx+λ1q+1∫Ωu∞x+q+1dx.
Then,(71)c=limk⟶∞Jλuk=limk⟶∞Jλuk−1pJλ′uk,ukX0=limk⟶∞sn∫Ωuk−u∞x+ps∗dx+sn∫Ωu∞x+ps∗dx+λ1p−1q+1∫Ωukx+q+1dx=snb+sn∫Ωu∞x+ps∗dx+λ1p−1q+1∫Ωu∞x+q+1dx≥snSKn/ps+sn∫Ωu∞x+p∗dx+λ1p−1q+1∫Ωu∞x+q+1dx.
Moreover, by Ho¨lder inequality, we have
(72)∫Ωu∞x+q+1dx≤Ωps∗−q+1/ps∗∫Ωu∞x+ps∗dxq+1/ps∗.
Thus,(73)c≥snSKn/ps+sn∫Ωu∞x+ps∗dx+λ1p−1q+1Ωps∗−q+1/ps∗∫Ωu∞x+ps∗dxq+1/ps≔snSKn/ps+hη,
where(74)hη=snηps∗+λ1p−1q+1Ωps∗−q+1/ps∗ηq+1withη=∫Ωu∞x+ps∗dx1/ps∗.
So,hη attains its minimum at η0=λp−q−1n−ps/2ps1/ps★−q+1Ω1/ps★ and
(75)hη0=−2n−n−psq+1p−q−12pq+1p−1−qn−2s2psq+1/ps★−q+1∣Ω∣λps∗/ps∗−q+1=−Mλps∗/ps∗−q+1.
Therefore,(76)c≥snSKn/ps−Mλps∗/ps∗−q+1,which is a contradiction. Therefore, b=0 and we obtain that uk⟶u∞ strongly in X0.Proposition 12.
There existsλ2>0 and u0∈X0 such that
(77)supt>0Jλtu0<snSKn/ps−Mλps∗/ps∗−q+1,
forλ∈0,λ2. In particular
(78)α~λ−<snSKn/ps−Mλps∗/ps∗−q+1.Proof.
We suppose there existsλ∗∗>0 such that s/nSKn/ps−Mλps∗/ps★−q+1>0 for all λ∈0,λ∗∗. By condition (17) we have that there is u0∈X0\0 such that
(79)Jλtu0≤supt≥0IK,ps∗tu0−λtq+1q+1∫Ωu0+q+1dx<snSKn/ps−λt0q+1q+1∫Ωu0+q+1dx.
Letλ∗∗∗≔t0q+1∫Ωu0+q+1dx/Mq+1ps∗−q+1/q+1. Therefore, for λ∈0,λ∗∗∗, we obtain that
(80)−t0q+1q+1λ∫Ωu0+q+1dx<−Mλps∗/ps∗−q+1.
Then, we have (77) holds.
Finally, letλ2=minλ∗,λ∗∗,λ∗∗∗, we obtain that
(81)α~λ−<snSKn/ps−Mλps∗/ps∗−q+1,for λ∈0,λ2 by the nature of fibering maps φut=Jλtu.Proof of Theorem 13.
There exist two sequencesuk+ and uk− in X0 such that
(82)Jλuk+⟶α~λ+,Jλ′uk+⟶0andJλuk−⟶α~λ−,Jλ′uk−⟶0,as k⟶∞ because of Propositions 11 and 12. From related properties of fibering maps φut, we have α~λ+<0. Similar to the Proof of Theorem 10, problem (1) admits two solutions u~1∈Nλ+ and u~2∈Nl− in X0. Moreover, these two solutions are distinct since Nλ+∩Nλ−=∅.



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