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

Abstract

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 $u\in H^1({\mathbb{R}}^3)$. 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.

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