Abstract: In the spirit of the classical work of P. H. Rabinowitz [ZAMP '92] we will discuss about existence
of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system
| `\{` | `-\Delta u+u+
\rho(x) \phi u=|u|^{p-1}u, \quad x \in
\mathbb{R}^3,` `-\Delta \phi=
\rho(x)u^2, \quad x \in
\mathbb{R}^3,` |
under different assumptions on `\rho: \mathbb{R}^3 \to \mathbb{R}_+`
at infinity. A singular perturbation of the above problem will be also
considered in various functional settings which are suitable for both
variational and perturbation methods.