JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/21972072197207Research ArticleExistence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearityhttps://orcid.org/0000-0003-0650-9556ZhangJiafengGuoWeiChuChangmuSuoHongminInfanteGennaroSchool of Data Science and Information EngineeringGuizhou Minzu UniversityGuiyang 550025Chinagzmu.edu.cn202028820202020020520202707202028820202020Copyright © 2020 Jiafeng Zhang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

Young Science and Technology Scholars of Guizhou Provincial Department of EducationKY164Science and Technology Foundation of Guizhou ProvinceQKH1084Guizhou UniversityKY479National Natural Science Foundation of China1186102111661021
1. Introduction and Main Result

The Schrödinger–Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field. Since the introduction of the Schrödinger–Poisson system by Benci and Fortunato , it has been extensively studied. For more detailed information, we refer the interested readers to [2, 3] and the references therein.

In recent years, some researchers extensively studied the Schrödinger–Poisson system with critical growth in an unbounded domain and obtained interesting results under various suitable assumptions (see, e.g., ).

But there are currently only a few results for the following Schrödinger–Poisson systems with critical nonlocal terms in a bounded domain : (1)Δu=qηϕu23u+gx,u,inΩ,Δϕ=qu21,inΩ,ϕ=u=0,onΩ,where Ωn;N3,q,η are real numbers; and g is a continuous function satisfying some suitable assumptions. In , assuming that η=1 and N=3,gx,u=λu, the author proved that system (1) has a positive ground state solution for any q>0 and λ3/10λ1,λ1, where λ is a real number and λ1 is the first eigenvalue of Δ. Later, when q=1,η=1,gx,u=λu, where λ is a real number, the authors in  studied system (1); they proved existence and nonexistence results of positive solutions when N=3 and existence of solutions in both the resonance and the nonresonance case for higher dimensions. For the case q=1,gx,u=λfx,λ>0 is a real number, f0, and fL2/21N, in ; when η=1, authors proved that system (1) has at least two positive solutions if 0<λ<λ for some λ>0 small enough, and when η=1, system (1) has at least one positive solution for any λ>0.

On the basis of the above literature, this paper continues to study system (1), and intends to deal with the following Schrödinger–Poisson system with critical nonlocal term and general nonlinearity that without (AR) condition: (2)Δu=ϕu23u+gx,u,inΩ,Δϕ=u21,inΩ,ϕ=u=0,onΩ,where ΩNN3 is an open bounded domain with smooth boundary Ω,

2=2N/N2 is the critical Sobolev exponent, and gCΩ×,,Gx,t=0tgx,sds..

Throughout this paper, we make the following assumptions:

(g1) gCΩ×,, and there exist constants c1,c2>0 with c1 which is small enough and p2,2 such that gx,tc1t+c2tp1,x,tΩ×.

(g2) There exists a constant K>0 big enough such that Gx,tKt2 for any xΩandt>0 large enough.

(g3) There are constants ρ>2 and ν>0 such that ρGx,tgx,tt+vt2,x,tΩ×.

The main difficulties in the present paper are to estimate the critical value and prove the boundedness of (PS) sequence due to the lack of compactness. In order to overcome the above difficulties, by analytic techniques, we shall give the estimate of critical value of associated functional so that system (2) has at least two nontrivial solutions.

Throughout this paper, we use the following notations:

The space H01Ω has the inner product u,v=Ωuvdxand the norm u2=Ωu2dx, and the norm in LpΩ is denoted by up=Ωupdx1/p

Br (respectively, Br) denotes the closed ball (respectively, the sphere) of center zero and radius r, i.e., Br=uH01Ω: ur,Br=uH01Ω: u=r

C,C0,C1,C2,... denote various positive constants, which may vary from line to line

Define the best constant S=infu2:uH01ΩΩu2dx=1, which is attained by the functions yεx=Cε/ε+x2N2/2 for all ε>0, where Cε=NN2εN2/4.

Theorem 1.

Assume that N3,g satisfies (g1), (g2), and (g3). Then, system (2) possesses at least two distinct nontrivial function pair solutions.

Remark 2.

Relative to [11, 12], the nonlinearity g is of a pure power form in [11, 12], and in the present paper, it is a general nonlinearity. Hence, we make a substantial improvement on the works of the former.

2. Proof of Main Result

Before proving our Theorem 1, we need the following lemmas.

Lemma 3 ([<xref ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B13">13</xref>]).

For every fixed uH01Ω, there exists a unique ϕuH01Ω that solves the second equation of (2), and

ϕu0a.e.inΩ

For all t>0,ϕtu=t21ϕu

ϕuS2/2u21

Ωu2dx=Ωϕuudx1/2ϕu2+1/2u2

Hence, according to the standard arguments as those in , system (2) can be converted into the following boundary value problem: (3)Δu=ϕuu23u+gx,u,inΩ,u=0,onΩ.

In order to study the existence of nontrivial solutions to problem (3), we shall firstly consider the existence of nontrivial solutions of the following problem: (4)Δu=ϕu+u+23+g+x,u,inΩ,u=0,onΩ,where (5)u+=maxu,0,g+x,t=gx,t,t0,0,t<0.

The energy functional corresponding to (4) is (6)Ju=12u21221Ωϕu+u+21dxΩG+x,udx,uH01Ω,where (7)G+x,t=0tg+x,sds.

J is well defined with JC1H01Ω, and (8)Ju,v=u,vΩϕu+u+22vdxΩg+x,uvdx,u,vH01Ω.

The critical points of the functional J are just weak solutions of problem (4). Let Uεx=yεx/Cε define a cutoff function φC0Ω such that (9)φx=1,xR,0,x2R,where B2R0Ω,0φx1for R<x<2R.

Put uεx=φxUεx,vεx=uεx/Ωuε2dx1/2; hence, Ωvε2dx=1.

Lemma 4 ([<xref ref-type="bibr" rid="B14">14</xref>, <xref ref-type="bibr" rid="B15">15</xref>]).

vεx satisfies the following estimates: (10)vε2=S+OεN2/2,(11)Ωvεqdx=OεN2q/4,1q<NN2,OεN/4lnε,q=NN2,Oε2NN2q/4,NN2<q<2.

Lemma 5.

Assume (g1) and (g3) hold; let unH01Ω be a sequence such that Junc,Jun0, wherec0,2/N+2SN/2. Then, there existsuH01Ω such that unu, up to a subsequence. Ju=0 and u is a nontrivial solution of problem (4).

Proof.

First, we prove that un is bounded in H01Ω. To prove the boundedness of un, arguing by contradiction, suppose that un. Set vn=un/un. Then, vn=1 and vnqC for 1q2. By (g3), we have (12)c+1+o1unJun1θJun,un=121θun2+1θ1221Ωϕun+un+21dx+1θΩg+x,unundxΩG+x,undxθ22θun2vθun22=un2θ22θvθun22,where θ=min2,ρ, which implies (13)12vθ2liminfnvn22.

Passing to a subsequence, we may assume that vnv in H01Ω, then vnv in LqΩ,1q<2, and vnva.e. in Ω. Hence, it follows from (13) that v0, and (14)0=limnc+o1un2=limnJunun2=limn121221un2Ωϕun+un+21dxΩG+x,unun2vn2dxlimn12KxΩ:vxσvn2dx=12KΩv2dx<0,where σ>0 is chosen such that xΩ:vxσ>0 and K is sufficiently big constant, which is a contradiction. Thus, un is bounded in H01Ω and there exists uH01Ω such that unu, up to a subsequence. Furthermore, Ju=0 by the weak continuity of J. If u=0 in Ω, since the term gx,u is subcritical, then Jun,un=o1 implies (15)un2Ωϕun+un+21dx=o1.

By Lemma 3-(iii), one has (16)ϕun+S2/2un+21.

It follows from (15) and (16) that (17)o1un21S2un222.

If un0, it contradicts c>0. Therefore, (18)un2SN/2+o1.

By (15) and (18), we get (19)Jun=12un21221Ωϕun+un+21dx+o1=2N+2un2+o12N+2SN/2+o1,which contradicts c<2/N+2SN/2. Thus, u0 and it is a nontrivial solution of problem (4).

Lemma 6.

Assume that g satisfies (g1) and (g2). Then, for ε>0 small enough, supt0Jtvε<2/N+2SN/2.

Proof.

For t0, we consider the functions (20)htJtvε=t22vε2t221221Ωϕvεvε21dxΩG+x,tvεdxNt2N+2vε2t221ΩG+x,tvεdx,h¯tNt2N+2vε2t221,where the above inequality comes from Lemma 3-(iv).

Notice that limt+ht=,h0=0, and ht>0 as t is sufficiently small. Therefore, supt0ht>0 is attained for some tε>0. Since (21)0=htε=2NtεN+2vε22tε2121Ωg+x,tvεvεdx,we have (22)vε2=tε22+N+22NtεΩg+x,tvεvεdxtε22,where the nonnegativity of Ωg+x,tvεvεdx comes from (g2), (g3), and the definition of g+. Hence, (23)tεvε2/22tε0,

It follows from (g1) that (24)g+x,tc1t+c2tp1.

Hence, we have (25)vε2tε22+c12Ωvε2dx+c2ptεp2Ωvε2dx.

By (10), (11), and (25), when ε is sufficiently small, we have tε22βS with β0,1.

On the other hand, the function h¯t attains its maximum at tε0=vε2/22 and it is increasing in the interval 0,tε0. By (10), (25), and G+x,tKt2 for t0, we deduce that (26)htεh¯tε0ΩG+x,tεvεdx2N+2vεNΩG+x,tεvεdx2N+2vεNKΩtε2vε2dx2N+2SN/2+OεN2/2KS2/22Ωvε2dx.

From (11) and the fact that K is sufficiently large, by choosing sufficiently small ε, we can obtain (27)supt0Jtvε=htε<2N+2SN/2.

Proof of Theorem 1.

It follows from (g1) that (28)g+x,tc1t+c2tp1,(29)G+x,t12c1t2+1pc2tp,for all t and xΩ. By the Sobolev inequality, (28) and (29), for c1 small enough, one has (30)Ju=12u21221Ωϕun+un+21dxΩG+x,udx12u21221S2u221C2u22Cpupp1C2u21221S2u221CpuP.

So, when r>0 is sufficiently small, there is α>0 such that Juα>0 for uBr0. Moreover, by the nonnegativity of G+x,u, for u0H01Ω\0, it holds that (31)Jtu0=t22u02t221221Ωϕu0u021dxΩG+x,tu0t22u02t221221Ωϕu0u021dx,limtJtu0,ast.

Hence, we can choose t0>0 such that t0u0>r and Jt0u00. Using the Mountain Pass Lemma, there is a sequence unH01Ω satisfying (32)Junca,Jun0,where (33)c=infγΓmaxt0,1Jγt,Γ=γtC0,1,H01Ωγ0=0,Jγ1<0.

According to Lemma 5 and Lemma 6, we can get a PSc sequence unH01Ω, and uH01Ω such that Ju=0. Thus, u is a solution of problem (4). And then, Ju,u=0, where u=minu,0. Hence, u=0, that is, u0. We conclude from the strong maximum principle that u is a positive solution of problem (3).

Next, we give the proof of two nontrivial solutions to system (2).

From the above discussion, problem (3) has a positive solution u1. Put kx,t=gx,tfort. Note that if u1 is a solution of (3), then, u1 is a solution of (3) replacing g with k. Hence, the equation (34)Δu=ϕuu23u+kx,u,inΩ,u=0,onΩhas at least a positive solution v. Let u2=v; then, u2 is a solution of (3).

Obviously u10,u20. So, problem (3) has at least two distinct nontrivial solutions u1 and u1; therefore, similar to [4, 5], system (2) has at least two distinct nontrivial function pair solutions u1,ϕu1 and u2,ϕu2.

Data Availability

The findings in this research do not make use of data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 11661021 and 11861021); the Science and Technology Foundation of Guizhou Province (No. QKH1084); the Young Science and Technology Scholars of Guizhou Provincial Department of Education (No. KY164); and the Key Laboratory of Advanced Manufacturing Technology, Ministry of Education, Guizhou University (No. KY479).

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