# Critical Fractionalp-Laplacian System with Negative Exponents

Authors: Qinghao Zhu; Jianming Qi
Journal: Journal of Function Spaces
 (2023)

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## Abstract

In this paper, we consider a class of fractionalp-Laplacian problems with critical and negative exponents. By decomposition of the Nehari manifold, the existence and multiplicity of nontrivial solutions for the above problems are established with respect to a sufficiently small parameter.


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## Body

## 1. Introduction and Main Result

Laplace transformation is an integral transformation commonly used in engineering mathematics. The transformation is a linear transformation that transforms a function with a real parameter numbertt≥0 into a function with complex parameters. The Laplace transform has extensive applications in many fields of engineering technology and scientific research, especially when it plays an important role in mechanical systems, electrical systems, automatic control systems, reliability systems, and random service systems. In circuit analysis, it is often necessary to solve the differential equation or the integral equation, which can be solved by the Laplace transformation. The application of nonlinear equations promoted the development of nonlinear sensitive electronic devices on the load side and grid side of the power system. The stable operation of the power system at each level can be effectually protected by exploring the nonlinear phenomena in the case of ferromagnetic resonance overvoltage situation. So, studying the Laplacian system is an important topic.In this paper, we study the following fractionalp-Laplacian system:
(1)−Δpsu=λαα+βuα−1vβ+fxu−γ,inΩ,−Δpsv=λβα+βuαvβ−1+gxv−γ,inΩ,u>0,v>0,inΩ,u=v=0,inℝN\Ω,where Ω is a bounded domain in ℝN, N>ps with s∈0,1, 0<γ<1<p, α>1,β>1 with α+β=ps∗, where ps∗=Np/N−ps is the fractional critical Sobolev exponent, and λ is a parameter. fx and gx satisfy the conditions f,g:Ω⟶ℝ such that 0<f,g∈Lγ∗Ω, where γ∗=ps∗/ps∗−1+γ. The fractional p-Laplacian operator −Δps is defined as
(2)−Δpsux=2limε↘0∫ℝN\Bεxuy−uxp−2uy−uxx−yN+psdy,x∈ℝN.We define(3)X=uu:ℝN⟶ℝismeasurable,uΩ∈LpΩ∫Qux−uypx−yN+psdxdy<+∞,where Q=ℝ2N\CΩ×CΩ with CΩ=ℝN\Ω and X0=u∈X∣u=0onCΩ. The space X is equipped with the norm uX=uLpΩ+uX0, where
(4)uX0p=∫Qux−uypx−yN+psdxdy.SetE=X0×X0 with the norm u,vp=uX0p+vX0p. u,v∈E is called a weak solution of problem (1) if
(5)∫Qux−uyp−2ux−uyφ1x−φ1yx−yN+psdxdy+∫Qvx−vyp−2vx−vyφ2x−φ2yx−yN+psdxdy=λαα+β∫Ωuα−1vβφ1dx+λβα+β∫Ωuαvβ−1φ2dx+∫Ωfxu−γφ1+gxv−γφ2dxfor all φ1,φ2∈E. The best fractional critical Sobolev constant S is defined as
(6)S=infu∈X0\0∫ℝ2Nux−uyp/x−yN+psdxdy∫ΩuxNp/N−psdxN−ps/N.In recent years, fractional Laplacian andp-Laplacian systems with subcritical and critical nonlinearities have been studied widely. Chen and Deng [1] and Li and Yang [2] studied the following critical fractional Laplacian system with a lower-order term:
(7)−Δsu=λuq−2u+2αα+βuα−2uvβ,inΩ,−Δsv=μvq−2v+2βα+βuαvβ−2v,inΩ,u=v=0,inℝN\Ω,where Ω⊂ℝN is a bounded domain with a smooth boundary, 0<s<1, 1<q<2, α,β>1 satisfy α+β=2s∗, where 2s∗=2N/N−2s and N>2s, α,β>0 are parameters. The main difficulty lies in finding the interval of c where the local PSc condition is satisfied. The authors both adopted the explicit formula of extremal function related to the best Sobolev constant and some useful estimates established in Barrios et al. [3] (Lemma 3.8) to overcome this difficulty. Compared with the fractional Laplacian system with critical nonlinearities, for the critical fractional p-Laplacian system with p≠2, we must face the difficulty that the explicit formula for minimizers of critical Sobolev constant S does not exist. Chen and Squassina [4] overcame this difficulty by borrowing the asymptotic estimates for minimizers of S, which were obtained in Brasco et al. [5].On the other hand, much attention has been focused on discussing the fractionalp-Laplacian system with negative exponent and subcritical nonlinearity. In [6], Goyal first investigated the following fractional Laplacian system with sign-changing nonlinearity:
(8)−Δsu=λfxu−q+αα+βbxuα−1vβ,inΩ,−Δsw=μgxw−q+βα+βbxuαwβ−1,inΩ,u>0,w>0,inΩ,u=w=0,inℝN\Ω,where 0<q<1, α>1, β>1, 2<α+β<2s∗−1 and bx is a sign-changing function. Using the decomposition of the Nehari manifold, the multiplicity of positive solutions for (8) with respect to the pair of parameters λ,μ was established. Furthermore, the author extended the above same result to the following p-fractional Laplacian system:
(9)−Δpsu=λfxu−q+αα+βbxuα−1vβ,inΩ,−Δpsw=μgxw−q+βα+βbxuαwβ−1,inΩ,u>0,w>0,inΩ,u=w=0,inℝN\Ω,where α>1, β>1, 0<q<1≤p−1<α+β<ps∗−1, bx is a sign-changing function. Very recently, Saoudi [7] investigated the following fractional p-Laplacian system:
(10)−Δpsu=λaxuq−2u+1−α2−α−βcxu−αv1−β,inΩ,−Δpsv=μbxvq−2v+1−β2−α−βcxu1−αv−β,inΩ,u=v=0,inℝN\Ω,where 0<α<1, 0<β<1, 2−α−β<p<q<ps∗. Using the variational method, the author proved that (10) has at least two positive solutions when the pair of parameters λ,μ satisfies certain conditions. We have found that it is easier to deal with subcritical problems than critical ones because compact conditions for Sobolev embedding are satisfied in the subcritical case, while global PCc condition for the energy functional corresponding to (1) does not usually hold in the case of critical problems. Moreover, Arora and Fiscella [8] studied a class of double-phase problems with a negative exponent and a critical Sobolev nonlinearity. The proof of the main result is based on a suitable minimization argument on the Nehari manifold. Zuo et al. [9] combined the effects of a nonlocal operator with critical nonlinearity. Suitable embedding results were developed to establish the existence of infinitely many solutions and to provide an estimate of the boundedness of these solutions. For more recent results on fractional p-Laplacian problems with singular terms and critical terms simultaneously, see [10–13] and references therein. In fact, Ghanmi et al. [10] studied a class of nonlocal p-Kirchhoff problems with a negative exponent and critical nonlinearity. The authors established the multiplicity of positive solutions to the above problems by using a truncation argument. Sang [11] considered a fractional critical system with p-Laplacian operator and negative exponents. By applying fibering map analysis, the existence of two positive solutions for the above systems was obtained. Furthermore, Saoudi et al. [12] proved the existence of solutions to a nonlocal problem with a singular term and a discontinuous critical nonlinearity. Fiscella et al. [13] investigated the existence of nontrivial solutions for critical systems driven by the fractional p-Laplacian operator. The main features of this paper are the presence of critical nonlinearities and singular terms.Motivated by the above results, we consider fractional criticalp-Laplacian system (1). We combine critical problems with negative exponents. Note that the energy functional corresponding to (1) is not differentiable in the sense of Gâteaux; the method used in [1, 2, 4, 14–20] cannot be applied to our problem directly. Since p≠2, we cannot also extend the methods used in a single equation with critical and negative exponents [21, 22] when p=2 to problem (1). We use the concentration compactness principle [23, 24] to avoid this barrier. Our idea comes from Wang et al. [25].The energy functional associated with problem (1) is defined by
(11)Iλu.v=1pu,vp−λα+β∫Ωuαvβdx−11−γ∫Ωfxu1−γ+gxv1−γdx.We define a set(12)Nλ=u,v∈Eu,vp−λ∫Ωuαvβdx−∫Ωfxu1−γ+gxv1−γdx=0and decompose Nλ with the following subsets:
(13)Nλ±=u,v∈Nλp−1+γu,vp−λα+β−1+γ∫Ωuαvβdx><0,Nλ0=u,v∈Nλp−1+γu,vp−λα+β−1+γ∫Ωuαvβdx=0.Our main result is the following.Theorem 1.
There existsλ∗>0 such that for every λ∈0,λ∗, problem (1) has at least two nontrivial solutions u1,v1 and u2,v2 in E. More specifically, u1,v1 is a local minimizer of Iλ in E with Iλu1,v1<0, and u2,v2∈Nλ− is a minimizer of Iλ on Nλ− with Iλu2,v2≥0.


## 2. Preliminaries and Some Lemmas

Let(14)Sα,β≔infu,v∈E\0,0u,vp∫Ωuαvβdxp/α+β,and the relationship betweenSα,β and S has been revealed in [4]. In order to prove our main result, the following lemmas are needed.Lemma 2.
The functionalIλ has a local minimum m in E with m<0.Proof.
Since(15)∫Qfxu1−γ+∫Ωgxv1−γdx≤fγ∗ups∗1−γ+gγ∗vps∗1−γ≤fγ∗uX0Sp1−γ+gγ∗uX0Sp1−γ≤fγ∗p/1+γ+gγ∗p/1+γ1+γ/puX0p+vX0pS1−γ/p=fγ∗p/1+γ+gγ∗p/1+γ1+γ/pu,vSp1−γ,(16)∫Ωuαvβdx≤u,vpSα,βα+β/p≤Sα,β−α+β/pu,vα+β,for every u,v∈E, where we have used the Hölder inequality, (6) and (14). We come to
(17)Iλu,v≥1pu,vp−λα+βSα,β−α+β/pu,vα+β−C1u,v1−γ,∀u,v∈E.
Hence, there existλ1>0 and R>0 such that Iλ is bounded on BR=u,v∈Eu,v≤R for every λ∈0,λ1. It follows that m=infu.v∈BRIλu,v is well defined for fixed λ∈0,λ1. Furthermore, choosing u,v∈BR with all u, v≠0, we have
(18)limt⟶0+Iλtu,tvt1−γ=−11−γ∫Ωfxu1−γ+gxv1−γdx<0;thus, Iλtu,tv<0 for all u, v≠0 and t small enough. Consequently, m<0.Lemma 3.
There existsu1,v1∈BR such that Iλu1,v1=m.Proof.
The definition ofm tells us that there exists a minimizing sequence un,vn such that limn⟶∞Iλun,vn=m<0. We assume that un≥0 and vn≥0. By un,vn≤R, there is a subsequence, which is still denoted by un,vn, such that
(19)un,vn⇀u1,v1inE.
In terms of the fractional concentration compactness principle [10] (Theorem 2.5), there exist two Borel regular measures μ and ν, J denumerable, xj∈Ω¯, μj≥0, νj≥0 with μj+νj>0,j∈J, such that
(20)∫ℝNunx−unypx−yN+psdy+∫ℝNvnx−vnypx−yN+psdy⇀dμ,unαvnβ⇀dν,(21)dμ≥∫ℝNu1x−u1ypx−yN+psdy+∫ℝNv1x−v1ypx−yN+psdy+∑j∈Jμjδxj,dν=u1αv1β+∑j∈Jνjδxj,μj≥Sνjp/ps∗.
It follows from Vitali’s theorem that(22)limn⟶∞∫ℝNfxun1−γ+gxvn1−γdx=∫ℝNfxu11−γ+gxv11−γdx,which, coupled with (20) and (21), gives
(23)m=limn⟶∞1pun,vnp−11−γ∫ℝNfxun1−γ+gxvn1−γdx−λα+β∫Ωunαvnβdx≥1p∫ℝN∫ℝNu1x−u1ypx−yN+psdydx+∫ℝN∫ℝNv1x−v1ypx−yN+psdydx+∑j∈JSνjp/ps∗−λα+β∫ℝNu1αv1βdx+∑j∈Jνj−11−γ∫ℝNfxu11−γ+gxv11−γdx.
IfJ=∅, then
(24)m≥1p∫ℝN∫ℝNu1x−u1ypx−yN+psdydx+∫ℝN∫ℝNv1x−v1ypx−yN+psdydx−λα+β∫ℝNu1αv1βdx−11−γ∫ℝNfxu11−γ+gxv11−γdx.
Combining with the definition ofm, we deduce that
(25)m=1p∫ℝN∫ℝNu1x−u1ypx−yN+psdydx+∫ℝN∫ℝNv1x−v1ypx−yN+psdydx−λα+β∫ℝNu1αv1βdx−11−γ∫ℝNfxu11−γ+gxv11−γdx.
In the following, we focus on showingJ=∅. Firstly, by (21), we have ∫ℝNdν≥∫ℝNu1αv1βdx, so we only need to prove that ∫ℝNdν=∫ℝNu1αv1βdx. We assume by contradiction that ∫ℝNdν>∫ℝNu1αv1βdx. Since ∑j∈Jνj=∫ℝNdν−∫ℝNu1αv1βdx>0, we derive that
(26)m≤m−1p∑j∈JSνjp/ps∗+λα+β∑j∈Jνj.
If0<νj<1, for all j∈J, it follows that
(27)m≤m−1p∑j∈JSνj+λα+β∑j∈Jνj=m+λα+β−Sp∑j∈Jνj,which is wrong if we choose λ<α+βS/p.
If there exists a subsequenceνjj∈1,2,⋯ such that νj≥1, then
(28)m≤m+λα+β−Sp∑j∈J\j∣νj≥1νj+λα+β∑νj≥1νj−1p∑νj≥1Sνjp/α+β<m,where λ<λ^2 with λ^2/α+β−S/p∑j∈J\j∣νj≥1νj+λ^2/α+β∑νj≥1νj<0, which is a contradiction. Set λ<λ2=minα+βS/p,λ^2, we have that J=∅. This completes the proof of Lemma 3.Lemma 4.
I is coercive in Λ.Proof.
Foru,v∈Λ, we have
(29)Iλu,v=1p−1α+βu,vp−11−γ−1α+β∫Ωfxu1−γ+gxv1−γdx≥1p−1α+βu,vp−11−γ−1α+βfγ∗p/1+γ+gγ∗p/1+γ1+γ/pSγ−1/pu,v1−γ,it follows from p>1−γ that Iλ is coercive on Λ.Let(30)λ3=p−1+γα+β−1+γSα,βα+β/pα+β−pα+β−1+γfγ∗p/1+γ+gγ∗p/1+γ−1+γ/pS1−γ/pα+β−p/γ+p−1.Lemma 5.
There exist two and only two numberst1 and t2 with 0<t1<t2 such that
(31)t1u,t1v∈Λ+,t2u,t2v∈Λ−,for all λ∈0,λ3 and u,v∈E.Proof.
Fort>0, we define ϕu,vt:0,+∞×E⟶ℝ by
(32)ϕu,vt=tp−α+βu,vp−λ∫Ωuαvβdx−t1−γ−α+β∫Ωfxu1−γ+gxv1−γdx.
Letϕ′u,vt=0, which yields
(33)t≔tmax=α+β−pu,vpα+β−1+γ∫Ωfxu1−γ+gxv1−γdx−1/p−1+γ,ϕ′′u,vtmax=u,vpp−α−βp−1+γα+β−pu,vpα+β−1+γ∫Ωfxu1−γ+gxv1−γdxp−α−β−2/1−γ−p<0.
Therefore,ϕu,vt achieves its maximum at tmax, and
(34)ϕu,vtmax=p−1+γα+β−1+γα+β−pα+β−1+γα+β−p/p−1+γu,vpα+β−1+γ/p−1+γ∫Ωfxu1−γ+gxv1−γdxα+β−p/p−1+γ−λ∫Ωuαvβdx>p−1+γα+β−1+γα+β−pα+β−1+γα+β−p/p−1+γu,vpα+β−1+γ/p−1+γfγ∗p/1+γ+gγ∗p/1+γ1+γ/pu,v/Sp1−γα+β−p/p−1+γ−λSα,β−α+β/pu,vα+β=p−1+γα+β−1+γα+β−pα+β−1+γα+β−p/p−1+γSp1−γα+β−p/p−1+γfγ∗p/1+γ+gγ∗p/1+γ−1+γα+β−p/pp−1+γ−λSα,β−α+β/pu,vα+β≔Eλu,vα+β.
Note thatEλ=0 if and only if λ=λ3. By the condition that λ<λ3, we deduce that Eλ>0 and ϕu,vtmax>0. A simple computation leads to
(35)limt⟶0+ϕ=∞,limt⟶+∞ϕ=−λ∫Ωuαvβdx<0.
Consequently,ϕu,vt has exactly two zero points t1 and t2 with 0<t1<t2 such that ϕ′t2<0<ϕ′t1.
Ifϕu,vt=0 and ϕu,vt>0, then
(36)tu,tvp−λ∫Ωtutvβdx−∫Ωfxtu1−γ+gxtv1−γdx=0,so tu,tv∈Λ.
Furthermore,ϕu,v′t>0 implies that
(37)p−α+βtp−α−β−1u,vp+γ+α+β−1t−γ−α−β∫Ωfxu1−γ+gxv1−γdx>0;by the definition of Λ, we have
(38)p−1+γtu,tvp−λα+β−1+γ∫Ωtuαtvβdx>0,namely, tu,tv∈Λ+. Therefore, t1u,t1v∈Λ+. Similarly, we get t2,u,t2v∈Λ−.Lemma 6.
Forλ∈0,λ3, we have Λ0=0,0.Proof.
Assume thatΛ0≠0,0. If u,v∈Λ0, then
(39)u,vp−λ∫Ωuαvβdx−∫Ωfxu1−γ+gxv1−γdx=0,(40)p−1+γu,vp−λα+β−1+γ∫Ωuαvβdx=0.
By (39) and (40), we get
(41)u,vp=α+β−1+γα+β−p∫Ωfxu1−γ+gxv1−γdx≤α+β−1+γα+β−pfγ∗p/1+γ+gγ∗p/1+γ1+γ/pS−1−γ/pu,v1−γ,(42)u,vp=λα+β−1+γp−1+γ∫Ωuαvβdx≤λα+β−1+γp−1+γSα,β−α+β/pu,vα+β.
Combining with (41) and (42), we derive that
(43)λ≥p−1+γα+β−1+γSα,βα+β/pu,vp−α−β≥p−1+γα+β−1+γSα,βα+β/pα+β−pα+β−1+γfγ∗p/1+γ+gγ∗p/1+γ−1+γ/pS1−γ/pα+β−p/γ+p−1=λ3,which contradicts with λ∈0,λ3.Lemma 7.
Assume thatλ∈0,λ3, then Λ− is a closed set in E-topology.Proof.
Assume thatun,vn⊂Λ− and un,vn⟶u,v strongly in E. In the following, we prove that u,v∈Λ−. For un,vn⊂Λ−, we have
(44)p−1+γun,vnp−λα+β−1+γ∫Ωunαvnβ<0,and so,
(45)p−1+γu,vp−λα+β−1+γ∫Ωuαvβ≤0.
Therefore,u,v∈Λ−∪Λ0. We can show that u,v∈Λ0 by contradiction. Otherwise, if u,v∈Λ0, we get that from the definition of Λ0 that
(46)u,vp−λ∫Ωuαvβdx−∫Ωfxu1−γ+gxv1−γdx=0,(47)p−1+γu,vp−λα+β−1+γ∫Ωuαvβdx=0.
By (34), (46), and (47), we deduce that
(48)Eλu,vα+β<p−1+γα+β−1+γα+β−pα+β−1+γα+β−p/p−1+γu,vpα+β−1+γ/p−1+γ∫Ωfxu1−γ+gxv1−γdxα+β−p/p−1+γ−λ∫Ωuαvβdx=p−1+γα+β−1+γα+β−pα+β−1+γα+β−p/p−1+γu,vpα+β−1+γ/p−1+γα+β−pu,vp/α+β−1+γα+β−p/p−1+γ−p−1+γα+β−1+γu,vp=0,which contradicts with Eλ>0 for all λ∈0,λ3.Lemma 8.
Letu,v∈Nλ−, then for every ϕ1,ϕ2∈E, there exist a number ε>0 and a continuous function h:0,+∞⟶0,+∞ such that
(49)h0=1,htu+tϕ1,v+tϕ2∈Nλ−,t<ε.Proof.
We defineF:E×ℝ+⟶ℝ as follows:
(50)Ft,r=rp−1+γu+tϕ1,v+tϕ2p−λrα+β−1+γ∫Ωu+tϕ1αv+tϕ2βdx−∫Ωfxu+tϕ11−γ+gxv+tϕ21−γdx.
Foru,v∈Nλ−⊂N, we have
(51)F0,1=u,vp−λ∫Ωuαvβdx−∫Ωfxu1−γ+gxv1−γdx=0,Fr0,1=p−1+γu,vp−λα+β−1+γ∫Ωuαvβdx<0.
We deduce that there existsε¯>0 such that Ft,r=0 has a unique continuous solution r=ht>0 for t<ε¯, t∈ℝ by using the implicit function theorem at the point 0,1. Since F0,1=0, then h0=1. By Ft,ht=0 for t<ε¯, t∈ℝ, we obtain
(52)0=htp−1+γu+tϕ1,v+tϕ2p−λhtα+β−1+γ∫Ωu+tϕ1αv+tϕ2βdx−∫Ωfxu+tϕ11−γ+gxv+tϕ21−γdx=1ht1−γhtu+tϕ1,v+tϕ2p−λ∫Ωhtu+tϕ1αhtv+tϕ2βdx−∫Ωfxhtu+tϕ11−γ+gxhtv+tϕ21−γdx,that is,
(53)htu+tϕ1,v+tϕ2∈Nλ,∀t∈ℝ,t<ε¯.
SinceFr0,1>0 and
(54)Frt,ht=p−1+γhtp−2+γu+tϕ1,v+tϕ2p−λα+β−1+γhtα+β−2+γ∫Ωu+tϕ1αv+tϕ2βdx=1ht2−γp−1+γhtu+tϕ1,v+tϕ2p−λα+β−1+γ∫Ωhtu+tϕ1αhtv+tϕ2βdx,we can choose ε>0 sufficiently small ε<ε¯ such that for every t∈ℝ with t<ε,
(55)p−1+γhtu+tϕ1,v+tϕ2p−λα+β−1+γ∫Ωhtu+tϕ1αhtv+tϕ2βdx<0,that is,
(56)htu+tϕ1,v+tϕ2∈Nλ−,∀t∈ℝ.
This completes the proof of Lemma8.Lemma 9.
There exists a constantλ4>0 such that Iλu,v≥0 for each u,v∈Nλ− and all λ∈0,λ4.Proof.
Assume that there existsu¯,v¯∈Nλ− such that Iλu¯,v¯<0; thus,
(57)λ1p−1α+β∫Ωu¯αv¯βdx−11−γ−1p∫Ωfxu¯1−γ+gxv¯1−γdx<0.
Equations (14) and (15) lead
(58)∫Ωu¯αv¯βdx<p−1+γα+βλ1−γα+β−pfγ∗p/1+γ+gγ∗p/1+γ1+γ/pSα,βSp1−γα+β/α+β−1+γ.
On the other hand, from (6) and (14), we have
(59)∫Ωuαvβdx≥p−1+γSα,βλα+β−1+γα+β/α+β−p,foreveryu,v∈Nλ−.
Choosing(60)λ<λ4=p−1+γα+β1−γα+β−pfγ∗p/1+γ+gγ∗p/1+γ1+γ/pSα,βSp1−γα+βα+β−p/1−γ−p·α+β−1+γp−1+γSα,βα+βα+β−1+γ/1−γ−p,we have
(61)∫Ωu¯αv¯βdx<p−1+γSα,βλα+β−1+γα+β/α+β−p,which is a contradiction. This completes the proof of Lemma 6.


## 3. The Proof of Theorem1

Our proof is divided into the following three steps.Step 1.
Problem (1) has a weak solution u2,v2 in E.Letm−=infu,v∈Nλ−Iλu,v>−∞. In terms of Lemma 4, we know that Iλ is coercive in Nλ; hence, m− is well defined. Ekeland’s variational principle guarantees to extract a minimizing sequence u~n,v~n⊂Nλ− with
(62)Iλu~n,v~n<m−+1n,Iλu~n,v~n≤Iλu,v+1nu−u~n,v−v~n,∀u,v∈Nλ−.We assume thatu~n, v~n≥0 in Ω and u~n,v~n (up to subsequence if necessary) converges to a nonnegative function, denoted by u2,v2 satisfying
(63)u~n,v~n⇀u2,v2weaklyinE,u~n,v~n⟶u2,v2a.e.inΩ.Fixϕ1,ϕ2∈E with ϕ1, ϕ2≥0, by using Lemma 5, there exists a sequence of functions fk:0,+∞⟶0,+∞ such that fk0=1 and fktu~k+tϕ1,v~k+tϕ2∈Nλ− for all u~k,v~k∈Nλ− and t small enough. It follows from the definition of Nλ that
(64)fkptu~k+tϕ1,v~k+tϕ2p−λfkα+βt∫Ωu~k+tϕ1αv~k+tϕ2βdx−fk1−γt∫Ωfxu~k+tϕ11−γ+gxv~k+tϕ21−γdx=0,(65)u~k,v~kp−λ∫Ωu~kαv~kβdx−∫Ωfxu~k1−γ+gxv~k1−γdx=0,so
(66)1kfkt−1u~k,v~k+tfktϕ1,ϕ2≥1kfktu~k+tϕ1,v~k+tϕ2−u~k,v~k≥Iu~k,v~k−Ifktu~k+tϕ1,v~k+tϕ2=1−fkptpu~k,v~kp+λfkα+βt−1α+β∫Ωu~k+tϕ1αv~k+tϕ2βdx+fk1−γt−11−γ∫Ωfxu~k+tϕ11−γ+gxv~k+tϕ21−γdx+fkptpu~k,v~kp−u~k+tϕ1,v~k+tϕ2p+λα+β∫Ωu~k+tϕ1αv~k+tϕ2βdx−∫Ωu~kαv~kβdx+11−γ∫Ωfxu~k+tϕ11−γ−u~k1−γdx+11−γ∫Ωgxv~k+tϕ21−γ−v~k1−γdx;thus,
(67)1kfk′0u~k,v~k+ϕ1,ϕ2≥−fk′0u~k,v~kp+λfk′0∫Ωu~kαv~kβdx+fk′0∫Ωfxu~k1−γ+gxv~k1−γdx−∫Q1x−yN+psu~kx−u~kyp−2u~kx−u~kyϕ1x−ϕ1y+v~kx−v~kyp−2v~kx−v~kyϕ2x−ϕ2ydxdy+λαα+β∫Ωu~kα−1v~kβϕ1dx+λβα+β∫Ωu~kαv~kβ−1ϕ2dx+∫Ωfxu~k−γϕ1+gxv~k−γϕ2dx=−fk′0u~k,v~kp−λ∫Ωu~kαv~kβdx−∫Ωfxu~k1−γ+gxv~k1−γdx−∫Q1x−yN+psu~kx−u~kyp−2u~kx−u~kyϕ1x−ϕ1y+v~kx−v~kyp−2v~kx−v~kyϕ2x−ϕ2ydxdy+λαα+β∫Ωu~kα−1v~kβϕ1dx+λβα+β∫Ωu~kαv~kβ−1ϕ2dx+∫Ωfxu~k−γϕ1+gxv~k−γϕ2dx.We derive that(68)∫Ωfxu~k−γϕ1+gxv~k−γϕ2dx≤1kfk′0u~k,v~k+ϕ1,ϕ2+∫Q1x−yN+psu~kx−u~kyp−2u~kx−u~kyϕ1x−ϕ1y+v~kx−v~kyp−2v~kx−v~kyϕ2x−ϕ2ydxdy−λαα+β∫Ωu~kα−1v~kβϕ1dx−λβα+β∫Ωu~kαv~kβ−1ϕ2dx.Furthermore, there exists a constantC2>0 such that fk′0≤C2. Taking k⟶∞, we deduce from Fatou’s lemma that
(69)∫Ωfxu2−γϕ1+gxv2−γϕ2dx≤liminfk⟶∞∫Ωfxu~k−γϕ1+gxv~k−γϕ2dx≤∫Q1x−yN+psu2x−u2yp−2u2x−u2yϕ1x−ϕ1y+v2x−v2yp−2v2x−v2yϕ2x−ϕ2ydxdy−λαα+β∫Ωu2α−1v2βϕ1dx−λβα+β∫Ωu2αv2β−1ϕ2dx.Consequently,(70)∫Q1x−yN+psu2x−u2yp−2u2x−u2yϕ1x−ϕ1y+v2x−v2yp−2v2x−v2yϕ2x−ϕ2ydxdy−λαα+β∫Ωu2α−1v2βϕ1dx−λβα+β∫Ωu2αv2β−1ϕ2dx−∫Ωfxu2−γϕ1+gxv2−γϕ2dx≥0,∀ϕ1,ϕ2∈CE.Sinceϕ1,ϕ2 is arbitrary, this inequality also holds for −ϕ1,ϕ2; thus,
(71)∫Q1x−yN+psu2x−u2yp−2u2x−u2yϕ1x−ϕ1y+v2x−v2yp−2v2x−v2yϕ2x−ϕ2ydxdy−λαα+β∫Ωu2α−1v2βϕ1dx−λβα+β∫Ωu2αv2β−1ϕ2dx−∫Ωfxu2−γϕ1+gxv2−γϕ2dx=0,∀ϕ1,ϕ2∈CE,which implies that u2,v2 is a weak solution of the problem (1).Step 2.
There exists a constantλ5>0 such that u2,v2∈Nλ− when 0<λ<λ5.For eachu,v∈Nλ−, we have
(72)Iλu,υ=1pu,υp−λα+β∫Ωuαυβdx−11−γu,υp−λ∫Ωuαυβdx=1p−11−γu,υp−λ1α+β−11−γ∫Ωuαυβdx<λα+β−1+γpp−1+γ∫Ωuαυβdx−λ1α+β−11−γ∫Ωuαυβdx=λα+β−1+γpp−1+γ−1α+β+11−γ∫Ωuαυβdx.We only need to prove thatm−=infu,v∈Nλ−Iλu,v<1/NSN/sp, when λ<λ~5 with
(73)λ~5α+β−1+γpp−1+γ−1α+β+11−γ∫Ωuαvβdx<1NSN/sp.Repeating the arguments as in Step1, we have
(74)∫Qu~kx−u~kypx−yN+psdxdy+∫Qv~kx−v~kypx−yN+psdxdy−λ∫Ωu~kαv~kβdx−∫Ωfxu~k1−γ+gxv~k1−γdx=o1.In the following, we show thatu2,v2∈Nλ−. Note that Nλ− is closed and u~k,v~k⇀u2,v2 in E, it suffices to prove that u~k,v~k⟶u2,v2. We suppose that u~k,v~k and u2,v2 satisfy the same properties as in (20) and (21). Let xj∈Ω in the support of ν and μ. Define φε∈Cc∞B2εxj with
(75)φε≥0,φεBε=1,φε∞≤1,∇φε∞≤C3ε.Applying (74), we derive that
(76)∫Qu~kx−u~kyp−2u~kx−u~kyϕ1x−ϕ1yx−yN+psdxdy+∫Qv~kx−v~kyp−2v~kx−v~kyϕ2x−ϕ2yx−yN+psdxdy−λαα+β∫Ωu~kα−1v~kβϕ1dx−λβα+β∫Ωu~kαv~kβ−1ϕ2dx−∫Ωfxu~k−γϕ1+gxv~k−γϕ2dx=o1k⟶∞,for every ϕ1,ϕ2∈E. Since φεu~k,φεv~k∈E, repeating the arguments as in Lemma 3, we derive that
(77)∫Qu~kx−u~kyp−2u~kx−u~kyφεu~kx−φεu~kyx−yN+psdxdy+∫Qv~kx−v~kyp−2v~kx−v~kyφεv~kx−φεv~kyx−yN+psdxdy−λαα+β∫Ωu~kα−1v~kβφεu~kdx−λβα+β∫Ωu~kαv~kβ−1φεv~kdx−∫Ωfxu~k−γφεu~k+gxv~k−γφεv~kdx=o1,k⟶∞. Moreover,
(78)∫Qφεdμ−λ∫Ωφεdν−∫Ωfxu21−γφε+gxv21−γφεdx=−∫Qu~kx−u~kyp−2u~kx−u~kyφεx−φεyu~kyx−yN+psdxdy−∫Qv~kx−v~kyp−2v~kx−v~kyφεx−φεyv~kyx−yN+psdxdy+o1k⟶∞.Thus,(79)0=limε⟶0∫Qφεdμ−λ∫Ωφεdν−∫Ωfxu21−γφε+gxv21−γφεdx=μi−λνj,by μj≥Sνjp/ps∗, namely, λνj≥Sνjp/ps∗, we get νj=0 or νj≥S/λN/sp.Next, we prove thatνj≥S/λcN/sp does not hold. Otherwise, there exists j0 such that νj0≥S/λN/sp, then
(80)m−=limk⟶∞1pu~k,v~kp−λα+β∫Ωu~kαv~kβdx−11−γ∫Ωfxu~k1−γ+gxv~k1−γdx≥λ11−γ−1α+β+1p−11−γα+β−1+γp−1+γ∫Ωu~kαv~kβdx≥λ11−γ−1α+β−11−γ−1pα+β−1+γp−1+γ∫Ωu2αv2βdx+∑j∈Jνj≥11−γ−1α+β−11−γ−1pα+β−1+γp−1+γλ1−N/spSN/sp≥1NSN/sp,where λ<λ¯5 with λ¯5<N1/1−γ−1/α+β−1/1−γ−1/pα+β−1+γ/p−1+γsp/N−sp; this contradicts the fact m−<1/NSN/ps. Choosing λ5=minλ~5,λ¯5, we have u~kαv~kβ⇀u2αv2β and u2,v2∈Nλ− when 0<λ<λ5.Step 3.
u1,v1 is a nontrivial solution to problem (1).Letλ∗=minλi,i=1,2,3,4,5, then Lemmas 2–9 and Steps 1 and 2 hold for all 0<λ<λ∗. Using Lemma 3, we have
(81)mint∈ℝIλu1+tϕ1,v1+tϕ2=Iλu1+tϕ1,v1+tϕ2t=0=Iλu1,v1,∀ϕ1,ϕ2∈E.Hence,u1,v1 is a weak solution of (1). Since
(82)0≤Iλu1+tϕ1,v1+tϕ2−Iλu1,v1=1pu1+tϕ1,v1+tϕ2p−λα+β∫Ωu1+tϕ1αv1+tϕ2βdx−11−γ∫Ωfxu1+tϕ11−γ+gxv1+tϕ21−γdx−1pu1,v1p+λα+β∫Ωu1αv1βdx+11−γ∫Ωfxu11−γ+gxv11−γdx≤1pu1+tϕ1,v1+tϕ2p−1pu1,v1p,ϕ1≥0,ϕ2≥0.Dividing byt>0 and passing to the limit as t⟶0, we obtain
(83)∫Qu1x−u1yp−2u1x−u1yϕ1x−ϕ1yx−yN+psdxdy+∫Qv1x−v1yp−2v1x−v1yϕ2x−ϕ2yx−yN+psdxdy≥0.Hence,u1,v1 is a nontrivial solution of (1). This completes the proof of Theorem 1.



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*Source: 9014456-2023-01-18.xml*