An infinite sequence of localized nodal solutions for Schr\"odinger-Poisson system with double potentials

TL;DR

This paper proves the existence of infinitely many localized, sign-changing solutions for a Schrödinger-Poisson system with double potentials, using variational methods and concentration analysis, extending previous results to more general settings.

Contribution

It introduces a novel approach combining penalization and invariant set methods to establish an infinite sequence of sign-changing solutions for Schrödinger-Poisson systems with double potentials.

Findings

Established existence of infinitely many localized sign-changing solutions.

Identified concentration points near potential minima or maxima.

Extended results to general nonlinear Schrödinger equations.

Abstract

In this paper, we study the existence of localized sign-changing (or nodal) solutions for the following nonlinear Schr\"odinger-Poisson system \begin{equation*} \begin{cases} -\varepsilon^2 \Delta u+V(x)u+\phi u=K(x)f(u),&\text{in}~\mathbb{R}^3,\\ -\varepsilon^2 \Delta \phi=u^2,&\text{in}~ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is small parameters, the linear potential $V$ and nonlinear potential $K$ are bounded and bounded away from zero. By using the penalization method together with the method of invariant sets of descending flow, we establish the existence of an infinite sequence of localized sign-changing solutions which are higher topological type solutions given by the minimax characterization of the symmetric mountain pass theorem and we determine a concrete set as the concentration position of these sign-changing solutions. For single potential, that is, linear potential $V$ or nonlinear potential $K$ is a positive constant, we prove that these localized sign-changing solutions concentrated near a local minimum set of the potential $V$ or a local maximum set of the potential $K$. Moreover, our method is works for the following nonlinear Schr\"odinger equation \begin{equation*} -\varepsilon^2 \Delta u+V(x)u=K(x)f(u),~\text{in}~\mathbb{R}^N \end{equation*} where $N\geq 2$. The result generalizes the result by Chen and Wang (Calc.Var.Partial Differential Equations 56:1-26, 2017).