In this paper we prove the existence of a nontrivial solution to the nonlinear Schrödinger–Maxwell equations in \mathbb{R}^{3} , assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions. Résumé: Dans cet article on démontre l'existence d'une solution non-banale et positive pour les équations non-linéaires de Schrödinger–Maxwell dans \mathbb{R}^{3} en supposant que le terme non-linéaire satisfait les hypothèses introduites par Berestycki et Lions.

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.