Elsevier

Nonlinear Analysis

Volume 198, September 2020, 111896
Nonlinear Analysis

Localized nodal solutions of higher topological type for nonlinear Schrödinger–Poisson system

https://doi.org/10.1016/j.na.2020.111896Get rights and content

Abstract

In this paper, we focus our attention on following Schrödinger–Poisson system: ε2Δu+V(x)u+λψu=|u|p2u,xR3,ε2Δψ=u2,uH1(R3).where 4<p<6 and ε,λ>0 are small positive parameters. Under a local condition imposed on the potential V, we study above system and obtain an infinite sequence of localized sign-changing solutions by applying the symmetric mountain pass theorem. Precisely, these solutions are constructed as higher topological type critical points of the energy functional and concentrated at a local minimum set of the potential V. Our method follows the same spirit of Chen and Wang’s method (Chen and Wang, 2017) which does not need any non-degeneracy condition of the limiting equations, but we cannot use it directly due to the presence of nonlocal term ψ(x)=14πR3u2(y)|xy|dy. We employ some new analytical skills to overcome the obstacles caused by the nonlocal term, our results improve and extend some related ones in the literature.

Section snippets

Introduction and main results

Consider following nonlinear Schrödinger–Poisson system ε2Δu+V(x)u+λψu=|u|p2u,xR3,ε2Δψ=u2,uH1(R3),where 4<p<6 and ε,λ>0 are small positive parameters. Principal aim of this paper is to construct k pairs of sign-changing solutions of system (1.1) for any positive integer k, and study the concentration phenomena of these semiclassical states as ε0. The potential V satisfies following assumption:

  • (V1)

    VC1(R3,R) and there exist b>a>0 such that aV(x)b,xR3;

  • (V2)

    there exists a bounded domain MR3 with

Preliminaries

In this section, we first recall some properties of function ψu, then introduce a new minimax theorem in the presence of invariant sets of negative gradient flow from [31].

To investigate (1.1), we will focus our attention on the following equivalent system by making the change of variable εy=x, Δu+V(εx)u+λψu=|u|p2u,xR3,Δψ=u2,uH1(R3),and the corresponding energy functional is Iε(u)=12R3|u|2+V(εx)u2dx+14λR3ψuu2dx1pR3|u|pdx,uH1(R3),where ψuD1,2(R3) is the unique solution of Δψu=u2

The Palais–Smale condition for penalized functional

To deal with the compactness issue of Palais–Smale sequence for functional Iε, we apply the penalization method employed by Byeon and Wang [11].

For any ε>0 and any set BR3, we set Bε={xR3|εxB}.Let ζC(R) be a cut-off function so that 0ζ(t)1 and ζ(t)0 for every tR, ζ(t)=0 if t0, ζ(t)>0 if t>0 and ζ(t)=1 if t1. Define χε(x)=0,ifxMεε6ζ(dist(x,Mε)),ifxMε.For ε small, one can verify that χε is a C1 function. For 2<2β<p, we define Qε(u)=(R3χεu2dx1)+β,uH1(R3),where (t)+=max{t,0}.

Existence of multiple sign-changing critical points of Γε

In this section, we will apply the abstract minimax Theorem 2.1 to obtain multiple sign-changing critical points of the penalized functional Γε.

Define P+{uH1(R3)|u0}andP{uH1(R3)|u0}.For σ>0, set P+σ{uH1(R3)|distH1(u,P+)<σ}andPσ{uH1(R3)|distH1(u,P)<σ},where distH1(u,A)infvAuv for uH1(R3) and AH1(R3). It is easy to see that Pσ=P+σ. Then we take X=H1(R3),P=P+σ,J=Γε,andW=PσP+σin Definition 2.1 and Theorem 2.1. Clearly, W is an open and symmetric subset of H1(R3) and H1(R3)W

Proof of main results

In this section, we will prove our main theorems by careful observation and analysis on the asymptotic behaviors of critical point uj,ε for the penalized functional Γε, and detailed estimates on the local Pohozaev identity associated with uj,ε.

Lemma 5.1

For any kN and 0<ε<εk, there exist a positive constant ϱ depending only on a and p and a positive constant ηk independent of ε and λ such that ϱuj,εηkandQε(uj,ε)ηk,1jk.

Proof

By a standard argument, we can prove this lemma, see, for instance, [15, Lemma

Acknowledgments

This work was supported by the NSFC (Nos. 11801574, 11971485, 11601145, 11701173), Natural Science Foundation of Hunan Province (No. 2019JJ50788), Central South University Innovation-Driven Project for Young Scholars (No. 2019CX022), and Excellent youth project of Education Department of Hunan Province (18B342), PR China.

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