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 (u2 star 1/vertical bar x vertical bar) u = vertical bar u vertical bar(p-2)u, where u is an element of H-1 (R-3). We are concerned mostly with p is an element of (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, which could be useful in other frameworks. The radial and nonradial cases turn out to yield essentially different situations.