Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger-Poisson systems

Abstract

We study a nonlinear Schrödinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- Δu+ u + λ^2 \left(\frac{1}{ω|x|^{N-2}}\star ρu^2\right) ρ(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \] 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 $2^{\ast }=2N/(N-2)$ 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.

43 pages