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

TL;DR

This paper proves the existence of solutions to a class of nonlinear Schr"odinger-Poisson systems with nonradial charge density, exploring conditions for solution concentration and addressing challenges due to lack of compactness.

Contribution

It establishes existence results for mountain-pass and least energy solutions under various conditions on the charge density, extending previous work to nonradial cases and unbounded domains.

Findings

Existence of mountain-pass solutions for the system.

Conditions for solution concentration at points.

Analysis of the problem for different charge density behaviors.

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.