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.

In this note we complete the study made in [1] on a Kirchhoff type equation with a Berestycki-Lions nonlinearity. We also correct Theorem 0.6 inside. * The author is 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 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 study a model which describes the relation of the matter\ud and the electromagnetic field from a unitarian standpoint in the spirit of the\ud ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear\ud perturbation of the Maxwell equation (SME). The particles are described by the\ud finite energy solitary waves of SME whose existence is due to the presence of\ud the nonlinearity. In the magnetostatic case (i.e. when the electric field E = 0 and\ud the magnetic field H does not depend on time) the semilinear Maxwell equations\ud reduce to the following semilinear equation\ud ∇×(∇×A) = f \ud (A) (1)\ud where “∇×” is the curl operator, f is the gradient of a smooth function f :R3→R\ud and A : R3 → R3 is the gauge potential related to the magnetic field H (H =\ud ∇×A). The presence of the curl operator causes (1) to be a strongly degenerateelliptic equation. The existence of a nontrivial finite energy solution of (1) having\ud a kind of cylindrical symmetry is proved. The proof is carried out by using a\ud variational approach based on two main ingredients: the Principle of symmetric\ud criticality of Palais, which allows to avoid the difficulties due to the curl operator,\ud and the concentration-compactness argument combined with a suitable minimization\ud argument