Revista Matemática Iberoamericana


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Published online first: 2020-01-07
DOI: 10.4171/rmi/1158

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

Carlo Mercuri[1] and Teresa Megan Tyler[2]

(1) Swansea University, UK
(2) Swansea University, UK

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 $$\left\{\begin{array}{lll} - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, &x\in \mathbb{R}^3, \\ -\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3, \end{array}\right.$$ under different assumptions on $\rho\colon \mathbb{R}^3\rightarrow \mathbb{R}_+$ 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 $$\left\{\begin{array}{lll} - \epsilon^2\Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, &x\in \mathbb{R}^3, \\ -\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3, \end{array}\right.$$ in various functional settings which are suitable for both variational and perturbation methods.

Keywords: Stationary nonlinear Schrödinger–Poisson system, weighted Sobolev spaces, Palais–Smale sequences, lack of compactness

Mercuri Carlo, Tyler Teresa Megan: On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density. Rev. Mat. Iberoam. Electronically published on January 7, 2020. doi: 10.4171/rmi/1158 (to appear in print)