Generalized Schrödinger-Poisson type systems
Antonio Azzollini, Pietro d’Avenia · Communications on Pure & Applied Analysis · 2012
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 boundeddomain, $1 0$, $f: R\to R$ is acontinuous 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 alsoconsidered and existence and nonexistence results are proved.