On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density

Abstract

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger–Poisson system

under different assumptions on $\rho :{\mathbb{R}}^3\to {\mathbb{R}}_{+}$ at infinity. Our results cover the range $p\in (2,3)$ where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a 'limiting problem' at infinity and of the possible unboundedness of the Palais–Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem

in various functional settings which are suitable for both variational and perturbation methods.