Semilinear elliptic equations and systems with measure data: existence and a priori estimates

Abstract

We give existence results and a priori estimates for a semilinear elliptic problem of the form \begin{equation*} \left\{ \begin{array}{l} -\Delta w=w^{Q}+\mu ,\qquad \text{in \thinspace }\Omega , \\ w=\lambda ,\qquad \qquad \qquad \quad \text{on }\partial \Omega , \end{array} \right. \end{equation*} where $Q>0,$ and $\mu $ and $\lambda $ are nonnegative Radon measures in $ \Omega $ and $\partial \Omega ,$ with $\int_{\Omega }\rho \,d\mu <+\infty ,$ where $\rho $ is the distance to $\partial \Omega .$ We extend the results to the case of systems \begin{equation*} \left\{ \begin{array}{l} -\Delta u=v^{p}+\mu ,\qquad -\Delta v=u^{q}+\eta ,\qquad \text{in }\Omega , \\ u=\lambda ,\qquad v=\kappa ,\qquad \text{on }\partial \Omega , \end{array} \right. \end{equation*} with $p,q>0,$ and the same assumptions on $\eta $ and $\kappa .$