Quantitative Properties on the Steady States to A Schr\"odinger-Poisson-Slater System

TL;DR

This paper provides a comprehensive analysis of the steady states of the Schrödinger-Poisson-Slater system, including existence, uniqueness, regularity, and asymptotic behavior, with the first known proof of uniqueness for this system.

Contribution

It offers the first proof of uniqueness for the steady states of the Schrödinger-Poisson-Slater system, along with detailed properties like existence and asymptotics.

Findings

Existence of steady states established.

Uniqueness of solutions proven for the first time.

Detailed regularity and asymptotic behavior analyzed.

Abstract

A relatively complete picture on the steady states of the following Schr$\ddot{o}$dinger-Poisson-Slater (SPS) system \[ \begin{cases} -\Delta Q+Q=VQ-C_{S}Q^{2}, & Q>0\text{ in }\mathbb{R}^{3}\\ Q(x)\to0, & \mbox{as }x\to\infty,\\ -\Delta V=Q^{2}, & \text{in }\mathbb{R}^{3}\\ V(x)\to0 & \mbox{as }x\to\infty. \end{cases} \] is given in this paper: existence, uniqueness, regularity and asymptotic behavior at infinity, where $C_{S}>0$ is a constant. To the author's knowledge, this is the first uniqueness result on SPS system.