Groundstates and infinitely many high energy solutions to a class of nonlinear Schr\"odinger-Poisson systems

TL;DR

This paper investigates the existence and multiplicity of solutions for a nonlinear Schr"odinger-Poisson system involving nonlocal terms, focusing on high energy solutions and ground states in dimensions 3 to 5.

Contribution

It establishes new existence and multiplicity results for solutions of a nonlinear Schr"odinger-Poisson system with nonlocal interactions, under conditions accommodating loss of compactness.

Findings

Existence of ground state solutions.

Infinitely many high energy solutions.

Results valid in dimensions 3, 4, and 5.

Abstract

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \] where $\omega = (N-2)|\mathbb{S}^{N-1}|,$ $\lambda>0,$ $q\in(2,2^{\ast} -1),$ $\rho:\mathbb R^N \to \mathbb R$ is nonnegative and locally bounded, $N=3,4,5$ and $2^*=2N/(N-2)$ is the critical Sobolev exponent. We prove existence and multiplicity of solutions working on a suitable finite energy space and under two separate assumptions which are compatible with instances where loss of compactness phenomena may occur.