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

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.