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 $u: \R^3 \to \R$ and $\mu>0$. The case $p <2$ being already studied, we consider here $p \geq 2$. 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 $p>2$.