On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density - Carlo Mercuri (Revista Matemática Iberoamericana, 2020)
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 a class of nonlinear Schrödinger-Poisson systems under different assumptions on a weight function at infinity. Our results cover a range of exponents, p,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 a singular perturbation of the same problem in various functional settings which are suitable for both variational and perturbation methods.