A Variational Analysis of a Gauged Nonlinear Schrödinger Equation

Abstract

This paper is motivated by a gauged Schrödinger 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.

21 pages, 1 figure