Generalized Schrödinger-Poisson type systems
Antonio Azzollini Pietro d’Avenia Valeria Luisi
In this paper we study the boundary value problem \begin{eqnarray*} -\Delta u+ \varepsilon q\Phi f(u)=\eta|u|^{p-1}u \quad in \quad \Omega, \\ - \Delta \Phi=2 qF(u) \quad in \quad \Omega, \\ u=\Phi=0 \quad on \quad \partial \Omega, \end{eqnarray*} where $\Omega \subset R^3$ is a smooth bounded domain, $1 < p < 5$, $\varepsilon,\eta= \pm 1$, $q>0$, $f: R\to R$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.
keywords: variational methods mountain pass. Schrödinger-Poisson equations
Klein-Gordon-Maxwell systems in a bounded domain
Pietro d’Avenia Lorenzo Pisani Gaetano Siciliano
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume homogeneous Dirichlet boundary conditions on $u$ and non homogeneous Neumann boundary conditions on $\phi$. In the "linear" case we prove the existence of a nontrivial solution when the coupling constant is sufficiently small. Moreover we show that a suitable nonlinear perturbation in the matter equation gives rise to infinitely many solutions. These problems have a variational structure so that we can apply global variational methods.
keywords: standing waves electrostatic field. Klein-Gordon-Maxwell system

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