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

TL;DR

This paper investigates the properties of minimizers for the Schrödinger-Poisson-Slater system, highlighting differences between radial and nonradial solutions, and provides a new lower bound for Coulomb energy relevant to the field.

Contribution

It introduces a general lower bound for Coulomb energy and explores the distinct behaviors of radial and nonradial minimizers in the system.

Findings

Radial minimizers exhibit specific behaviors influenced by the static case.

Nonradial minimizers display different properties, indicating diverse solution structures.

A new lower bound for Coulomb energy is established, applicable to broader contexts.

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