# Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

Authors: Jiafeng Zhang; Wei Guo; Changmu Chu; Hongmin Suo
Journal: Journal of Function Spaces
 (2020)

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## Abstract

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.


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## Body

## 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 [1], 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., [4–10]).But there are currently only a few results for the following Schrödinger–Poisson systems with critical nonlocal terms in a bounded domain [11–13]:
(1)−Δu=qηϕu2∗−3u+gx,u,inΩ,−Δϕ=qu2∗−1,inΩ,ϕ=u=0,on∂Ω,where Ω⊂ℝn;N≥3,q,η are real numbers; and g is a continuous function satisfying some suitable assumptions. In [11], 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 λ∈3/10λ1,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 [12] 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, f≥0, and f∈L2∗/2∗−1ℝN, in [13]; 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=ϕu2∗−3u+gx,u,inΩ,−Δϕ=u2∗−1,inΩ,ϕ=u=0,on∂Ω,where Ω⊂ℝNN≥3 is an open bounded domain with smooth boundary ∂Ω,2∗=2N/N−2 is the critical Sobolev exponent, and g∈CΩ×ℝ,ℝ,Gx,t=∫0tgx,sds..Throughout this paper, we make the following assumptions:(g1) g∈CΩ×ℝ,ℝ, and there exist constants c1,c2>0 with c1 which is small enough and p∈2,2∗ such that ∣gx,t∣≤c1∣t∣+c2tp−1,∀x,t∈Ω×ℝ.(g2) There exists a constant K>0 big enough such that Gx,t≥Kt2 for any x∈Ωand∣t∣>0 large enough.(g3) There are constants ρ>2 and ν>0 such that ρGx,t≤gx,tt+vt2,∀x,t∈Ω×ℝ.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:(i)
The spaceH01Ω has the inner product u,v=∫Ω∇u⋅∇vdxand the norm u2=∫Ω∇u2dx, and the norm in LpΩ is denoted by up=∫Ωupdx1/p(ii)
Br (respectively, ∂Br) denotes the closed ball (respectively, the sphere) of center zero and radius r, i.e., Br=u∈H01Ω:u≤r,∂Br=u∈H01Ω:u=r(iii)
C,C0,C1,C2,... denote various positive constants, which may vary from line to line(iv)
Define the best constantS=infu2:u∈H01Ω∫Ωu2∗dx=1, which is attained by the functions yεx=Cε/ε+x2N−2/2 for all ε>0, where Cε=NN−2εN−2/4.Theorem 1.
Assume thatN≥3,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 Theorem1, we need the following lemmas.Lemma 3 ([12, 13]).
For every fixedu∈H01Ω, there exists a unique ϕu∈H01Ω that solves the second equation of (2), and
(i)
ϕu≥0a.e.inΩ(ii)
For allt>0,ϕtu=t2∗−1ϕu(iii)
ϕu≤S−2∗/2u2∗−1(iv)
∫Ωu2∗dx=∫Ω∇ϕu∇udx≤1/2ϕu2+1/2u2Hence, according to the standard arguments as those in [1], system (2) can be converted into the following boundary value problem:
(3)−Δu=ϕuu2∗−3u+gx,u,inΩ,u=0,on∂Ω.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+2∗−3+g+x,u,inΩ,u=0,on∂Ω,where
(5)u+=maxu,0,g+x,t=gx,t,t≥0,0,t<0.The energy functional corresponding to (4) is
(6)Ju=12u2−122∗−1∫Ωϕu+u+2∗−1dx−∫ΩG+x,udx,u∈H01Ω,where
(7)G+x,t=∫0tg+x,sds.J is well defined with J∈C1H01Ω,ℝ and
(8)J′u,v=u,v−∫Ωϕu+u+2∗−2vdx−∫Ωg+x,uvdx,u,v∈H01Ω.The critical points of the functionalJ are just weak solutions of problem (4). Let Uεx=yεx/Cε define a cutoff function φ∈C0∞Ω such that
(9)φx=1,x≤R,0,x≥2R,where B2R0⊂Ω,0≤φx≤1for R<∣x∣<2R.Putuεx=φxUεx,vεx=uεx/∫Ωuε2∗dx1/2∗; hence, ∫Ωvε2∗dx=1.Lemma 4 ([14, 15]).
vεx satisfies the following estimates:
(10)vε2=S+OεN−2/2,(11)∫Ωvεqdx=OεN−2q/4,1≤q<NN−2,OεN/4lnε,q=NN−2,Oε2N−N−2q/4,NN−2<q<2∗.Lemma 5.
Assume (g1) and (g3) hold; let un⊂H01Ω be a sequence such that Jun⟶c,J′un⟶0, wherec∈0,2/N+2SN/2. Then, there existsu∈H01Ω such that un⟶u, up to a subsequence. J′u=0 and u is a nontrivial solution of problem (4).Proof.
First, we prove thatun is bounded in H01Ω. To prove the boundedness of un, arguing by contradiction, suppose that un⟶∞. Set vn=un/un. Then, vn=1 and vnq≤C for 1≤q≤2∗. By (g3), we have
(12)c+1+o1un≥Jun−1θJ′un,un=12−1θun2+1θ−122∗−1∫Ωϕun+un+2∗−1dx+1θ∫Ωg+x,unundx−∫ΩG+x,undx≥θ−22θun2−vθun22=un2θ−22θ−vθun22,where θ=min2∗,ρ, which implies
(13)1≤2vθ−2liminfn⟶∞vn22.Passing to a subsequence, we may assume thatvn⟶v in H01Ω, then vn⟶v in LqΩ,1≤q<2∗, and vn⟶va.e. in Ω. Hence, it follows from (13) that v≠0, and
(14)0=limn⟶∞c+o1un2=limn⟶∞Junun2=limn⟶∞12−122∗−1un2∫Ωϕun+un+2∗−1dx−∫ΩG+x,unun2vn2dx≤limn⟶∞12−K∫x∈Ω:vx≥σvn2dx=12−K∫Ωv2dx<0,where σ>0 is chosen such that x∈Ω:vx≥σ>0 and K is sufficiently big constant, which is a contradiction. Thus, un is bounded in H01Ω and there exists u∈H01Ω such that un⟶u, up to a subsequence. Furthermore, J′u=0 by the weak continuity of J′. If u=0 in Ω, since the term gx,u is subcritical, then J′un,un=o1 implies
(15)un2−∫Ωϕun+un+2∗−1dx=o1.By Lemma3-(iii), one has
(16)ϕun+≤S−2∗/2un+2∗−1.It follows from (15) and (16) that
(17)o1≥un21−S−2∗un22∗−2.Ifun⟶0, it contradicts c>0. Therefore,
(18)un2≥SN/2+o1.By (15) and (18), we get
(19)Jun=12un2−122∗−1∫Ωϕun+un+2∗−1dx+o1=2N+2un2+o1≥2N+2SN/2+o1,which contradicts c<2/N+2SN/2. Thus, u≠0 and it is a nontrivial solution of problem (4).Lemma 6.
Assume thatg satisfies (g1) and (g2). Then, for ε>0 small enough, supt≥0Jtvε<2/N+2SN/2.Proof.
Fort≥0, we consider the functions
(20)ht≔Jtvε=t22vε2−t22∗−122∗−1∫Ωϕvεvε2∗−1dx−∫ΩG+x,tvεdx≤Nt2N+2vε2−t2∗2∗−1−∫ΩG+x,tvεdx,h¯t≔Nt2N+2vε2−t2∗2∗−1,where the above inequality comes from Lemma 3-(iv).Notice thatlimt⟶+∞ht=−∞,h0=0, and ht>0 as t is sufficiently small. Therefore, supt≥0ht>0 is attained for some tε>0. Since
(21)0=h′tε=2NtεN+2vε2−2∗tε2∗−12∗−1−∫Ωg+x,tvεvεdx,we have
(22)vε2=tε2∗−2+N+22Ntε∫Ωg+x,tvεvεdx≥tε2∗−2,where the nonnegativity of ∫Ωg+x,tvεvεdx comes from (g2), (g3), and the definition of g+. Hence,
(23)tε≤vε2/2∗−2≜tε0,It follows from (g1) that
(24)g+x,t≤c1t+c2tp−1.Hence, we have(25)vε2≤tε2∗−2+c12∫Ωvε2dx+c2ptεp−2∫Ωvε2dx.By (10), (11), and (25), when ε is sufficiently small, we have tε2∗−2≥βS with β∈0,1.On the other hand, the functionh¯t attains its maximum at tε0=vε2/2∗−2 and it is increasing in the interval 0,tε0. By (10), (25), and G+x,t≥Kt2 for t≥0, we deduce that
(26)htε≤h¯tε0−∫ΩG+x,tεvεdx≤2N+2vεN−∫ΩG+x,tεvεdx≤2N+2vεN−K∫Ωtε2vε2dx≤2N+2SN/2+OεN−2/2−KS2/2∗−2∫Ωvε2dx.From (11) and the fact that K is sufficiently large, by choosing sufficiently small ε, we can obtain
(27)supt≥0Jtvε=htε<2N+2SN/2.Proof of Theorem 1.
It follows from (g1) that
(28)g+x,t≤c1t+c2tp−1,(29)G+x,t≤12c1t2+1pc2tp,for all t∈ℝ and x∈Ω. By the Sobolev inequality, (28) and (29), for c1 small enough, one has
(30)Ju=12u2−122∗−1∫Ωϕun+un+2∗−1dx−∫ΩG+x,udx≥12u2122∗−1S−2∗u22∗−1−C2u22−Cpupp≥1−C2u2−122∗−1S−2∗u22∗−1−CpuP.So, whenr>0 is sufficiently small, there is α>0 such that Ju≥α>0 for u∈∂Br0. Moreover, by the nonnegativity of G+x,u, for u0∈H01Ω\0, it holds that
(31)Jtu0=t22u02−t22∗−122∗−1∫Ωϕu0u02∗−1dx−∫ΩG+x,tu0≤t22u02−t22∗−122∗−1∫Ωϕu0u02∗−1dx,limt⟶∞Jtu0⟶−∞,ast⟶∞.Hence, we can chooset0>0 such that t0u0>r and Jt0u0≤0. Using the Mountain Pass Lemma, there is a sequence un⊂H01Ω satisfying
(32)Jun⟶c≥a,J′un⟶0,where
(33)c=infγ∈Γmaxt∈0,1Jγt,Γ=γt∈C0,1,H01Ωγ0=0,Jγ1<0.According to Lemma5 and Lemma 6, we can get a PSc sequence un⊂H01Ω, and u∈H01Ω such that J′u=0. Thus, u is a solution of problem (4). And then, J′u,u−=0, where u−=minu,0. Hence, u−=0, that is, u≥0. 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∈ℝ. 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=ϕuu2∗−3u+kx,u,inΩ,u=0,on∂Ωhas at least a positive solution v. Let u2=v; then, u2 is a solution of (3).Obviouslyu1≥0,u2≤0. 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.



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*Source: 2197207-2020-08-28.xml*