On a class of nonlinear Schr\"odinger-Poisson systems involving a nonradial charge density

Abstract

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schr\"odinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schr\"odinger-Poisson system \begin{equation}\nonumber \left\{\begin{array}{lll} - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, \qquad &x\in \mathbb R^3, \,\,\, -\Delta \phi=\rho(x) u^2,\ & x\in \mathbb R^3, \end{array} \right. \end{equation} under different assumptions on $\rho: \mathbb R^3\rightarrow \mathbb R_+$ 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 \begin{equation}\nonumber \left\{\begin{array}{lll} - \epsilon^2\Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, \qquad &x\in \mathbb R^3, \,\,\, -\Delta \phi=\rho(x) u^2,\ & x\in \mathbb R^3, \end{array} \right. \end{equation} in various functional settings which are suitable for both variational and perturbation methods.