On the Schrodinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases

This paper is motivated by the study of a version of the so-called Schrodinger-Poisson-Slater problem: $$ - \Delta u + \omega u + \lambda (u^2 \star \frac{1}{|x|}) u=|u|^{p-2}u,$$ where $u \in H^1(\R^3)$. We are concerned mostly with $p \in (2,3)$. The behavior of radial minimizers motivates the study of the static case $\omega=0$. Among other things, we obtain a general lower bound for the Coulomb energy, that could be useful in other frameworks. The radial and nonradial cases turn out to yield essentially different situations.