Ground and bound states for a static Schrodinger-Poisson-Slater problem

TL;DR

This paper investigates the existence and properties of solutions to a nonlinear Schrödinger-Poisson-Slater equation in three dimensions, focusing on the case where the nonlinearity exponent is greater than or equal to two, and establishes decay rates for radial solutions.

Contribution

It extends the analysis of the Schrödinger-Poisson-Slater problem to the case p ≥ 2, including existence results for ground and bound states, and characterizes decay behavior of radial solutions.

Findings

Existence of ground and bound states for p > 2

p=2 identified as a critical case

Radial solutions decay exponentially at infinity

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 $p>2$.