Alessio Pomponio

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.

Standing waves for a gauged nonlinear Schrödinger equation with a vortex point

This paper is motivated by a gauged Schrodinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.

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 Electrostatic Born–Infeld Equation with Extended Charges

AbstractIn this paper, we deal with the electrostatic Born–Infeld equation

Infinitely many positive solutions for a Schrödinger–Poisson system ☆

\left\{\begin{array}{ll}-\operatorname{div} \left(\displaystyle\frac{\nabla\phi}{\sqrt{1-|\nabla \phi|^2}} \right)= \rho \quad{in} \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty} \phi(x)= 0,\end{array}\right. \quad \quad \quad \quad ({\mathcal{BI}})

Vortex ground states for Klein-Gordon-Maxwell-Proca type systems

-div∇ϕ1-|∇ϕ|2=ρinRN,lim|x|→∞ϕ(x)=0,(BI)where