In this paper we study the nonlinear Schrödinger-Maxwell equationsIf V is a positive constant, we prove the existence of a ground state solution (u, φ) for 2 < p < 5. The non-constant potential case is treated for 3 < p < 5, and V possibly unbounded below. Existence and nonexistence results are proved also when the nonlinearity exhibits a critical growth.

Abstract. In this paper, we deal with the electrostatic Born-Infeld equationwhere ρ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential φ and finiteness of the energy of the electrostatic field −∇φ. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming ρ is radially distributed, we recover the weak formulation of (BI) and the regularity of the solution of the Poisson equation (under the same smootheness assumptions).In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if ρ is smooth. Then we analyze the case where the density ρ is a superposition of point charges and discuss the results in [17]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.

In this paper we prove the existence of a nontrivial solution to the nonlinear Schrödinger-Maxwell equations in R 3 , assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions. * The authors are supported by M.I.U.R. -P.R.I.N. "Metodi variazionali e topologici nello studio di fenomeni non lineari"

In this paper we prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrödinger-Maxwell system in R 3 and to the nonlinear elliptic Kirchhoff equation in R N assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.

In this paper we prove the existence of a positive solution to the equation −∆u + V (x)u = g(u) in R N , assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution.