The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system

TL;DR

This paper investigates the existence and behavior of solutions to a parameter-dependent Schrödinger-Poisson system with various potentials, introducing new methods that require only super-linear growth conditions on the nonlinearity.

Contribution

It provides new existence results for solutions under less restrictive conditions and introduces novel observations on Poisson equation solutions, applicable to related problems.

Findings

Existence of solutions for positive radial potentials via mountain pass theorem.

Existence of solutions for indefinite potentials using local linking and Morse theory.

Asymptotic behavior analysis of solutions.

Abstract

In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3, -\Delta\phi=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which means that the Schr\"{o}dinger operator $-\Delta +V$ is allowed to be indefinite. To deal with this more difficult case, by a local linking argument and Morse theory, the system has a nontrivial solution. Furthermore, we also show the asymptotical behavior result of this solution. Additionally, the proofs rely on new observations regarding the solutions of the Poisson equation. As a main novelty with respect to corresponding results in \cite{MR4527586,MR3148130,MR2810583}, we only assume that $f$ satisfies the super-linear growth condition at the origin. We believe that the methodology developed here can be adapted to study related problems concerning the existence of solutions for Schr\"{o}dinger-Poisson system.