Ground and bound states for a static Schrodinger-Poisson-Slater problem

Abstract

In this paper the following version of the Schrodinger-Poisson-Slater problem is studied: $$-\Delta u+(u^2\star \frac{1}{|4\pi x|})u=\mu |u{|}^{p-1}u,$$ where $u:{\mathbb{R}}^3\to \mathbb{R}$ and $\mu >0$. The case $p<2$ being already studied, we consider here $p\ge 2$. For $p>2$ we study both the existence of ground and bound states. It turns out that $p=2$ is critical in a certain sense, and will be studied separately. Finally, we prove that radial solutions satisfy a point-wise exponential decay at infinity for $p>2$.