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.
Categories: math.AP math-ph math.MP