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

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.