Asymptotic profile of ground states for the Schr\"{o}dinger-Poisson-Slater equation

TL;DR

This paper analyzes the asymptotic behavior of ground states for the Schrödinger-Poisson-Slater equation in three dimensions, revealing their limit profiles as the parameter  approaches infinity through variational methods.

Contribution

It provides a novel characterization of the limit profiles of positive ground states for large  using direct variational analysis and energy level comparisons.

Findings

Limit profiles of ground states are characterized as   

Ground state energy levels are compared to determine asymptotic behavior

The analysis applies for p in (3,6) and   

Abstract

We study the Schr\"{o}dinger-Poisson-Slater equation $$-\Delta u + u+\lambda(I_{2}*|u|^2)u=|u|^{p-2}u\quad\text{in $\mathbb R^3$},$$ where $p\in (3,6)$ and $\lambda>0$. By using direct variational analysis based on the comparison of the ground state energy levels, we obtain a characterization of the limit profile of the positive ground states for $\lambda\to \infty$.