In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein-Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

In this paper we obtain some non-existence results for the Klein-Gordon equation coupled with the electrostatic field. The method relies on the deduction of some suitable Pohožaev identity which provides necessary conditions to get existence of nontrivial solutions. The case of Maxwell-Schrödinger type coupled equations is also considered.

We study the semiclassical limit for the following system of Maxwell-Schrödinger equations:This system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field: the unknowns v and φ represent the wave function associated to the particle and the electric potential, respectively. By using localized energy method, we construct a family of positive radially symmetric bound states (v , φ ) such that v concentrates around a sphere {|x| = s 0 } when → 0.

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

We study the following system of Maxwell-Schrödinger equationswhere δ > 0, u, ψ : R N → R, f : R → R, N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δ K > 0 such that, for 0 < δ < δ K , the system has a solution (u δ , ψ δ ) with the property that u δ has K spikes centered at the points Q δ 1 , . . . , Q δ K . Furthermore, setting l δ = min i = j |Q δ i − Q δ j |, then, as δ → 0, ( 1 l δ Q δ 1 , . . . , 1 l δ Q δ K ) approaches an optimal configuration for the following maximization problem:Subject class: Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40