A Variational Analysis of a Gauged Nonlinear Schr\"odinger Equation

TL;DR

This paper analyzes a gauged nonlinear Schr"odinger equation with a Chern-Simons term, establishing conditions for the boundedness of the associated energy functional and proving existence or non-existence of positive solutions.

Contribution

It provides a variational analysis of the energy functional for the gauged Schr"odinger equation, identifying explicit thresholds for parameters that determine solution existence.

Findings

The energy functional can be bounded below depending on the parameter .

Explicit threshold value for  determines boundedness.

Existence and non-existence of positive solutions are established based on .

Abstract

This paper is motivated by a gauged Schr\"odinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \Delta u(x) + \left(\omega + \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds. $$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $\omega $. Quite surprisingly, the threshold value for $\omega $ is explicit. From this study we prove existence and non-existence of positive solutions.