Antonio Azzollini

Ground state solutions for the nonlinear Schrödinger-Maxwell equations

Abstract In this paper we study the nonlinear Schrodinger–Maxwell equations { − Δ u + V ( x ) u + ϕ u = | u | p − 1 u in R 3 , − Δ ϕ = u 2 in R 3 . If 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.

On the Schrödinger–Maxwell equations under the effect of a general nonlinear term☆

Abstract In this paper we prove the existence of a nontrivial solution to the nonlinear Schrodinger–Maxwell equations in R 3 , assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.

On a “zero mass” nonlinear Schrödinger equation

Abstract We look for positive solutions to the nonlinear Schrödinger equation −Ɛ2∆u − V (x)f′(u) = 0 in ℝN, where V is a continuous bounded positive potential and f satisfies particular growth conditions which make our problem fall in the so called “zero mass case”. We prove an existence result for any Ɛ > 0, and a multiplicity result for Ɛ sufficiently small.

On the Schrödinger–Born–Infeld System

In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrödinger and the Born–Infeld agrangians, this latter replacing the role that, classically, is played by the Maxwell Lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.