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 (Formula presented.) where (Formula presented.) This problem is the Euler-Lagrange equation of a certain energy functional. We study the global behavior of that functional. We show that for p ∈ (1, 3), the functional may be bounded from below or not, depending on ω. Quite surprisingly, the threshold value for ω is explicit. From this study we prove existence and non-existence of positive solutions.