A Variational Analysis of a Gauged Nonlinear Schrödinger Equation.

This paper is motivated by a gauged Schrodinger 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) + (omega + h(2)(vertical bar x vertical bar) /vertical bar x vertical bar(2) + integral(infinity)(vertical bar x vertical bar) h(s)/s u(2)(s) ds )u(x) = vertical bar u(x)vertical bar(p-1) u(x), where h(r) = 1/2 integral(r)(0) su(2)(s) ds. 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 is an element of (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 solutio