Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

Abstract

We prove existence and multiplicity results for the fractional Schroedinger--Poisson--Slater equation $(-\Delta {)}^{s}u+(I_{\alpha }\ast u^2)u=f(|x|,u)$ in ${\mathbb{R}}^N$, where $0<s<1$ and $\alpha \in (1,N)$. We seek solutions in a fractional Coulomb-Sobolev space and employ new tools in critical point theory that link the behavior of $f$ at zero and at infinity to the scaling properties of the left-hand side. For several regimes of $f$, we establish compactness for an associated action functional and obtain multiple solutions as critical points, with the number governed by the interaction of $f$ with a sequence of eigenvalues $\{{\lambda }_k\}$ defined via the ${\mathbb{Z}}_2$ cohomological index of Fadell and Rabinowitz (rather than the classical Krasnosel'skii genus). In this fractional setting we also prove new regularity results and necessary conditions for the existence of solutions.