Maximal Solutions of Semilinear Elliptic Equations with Locally Integrable Forcing Term

We study the existence of a maximal solution of $-\text{Gd}u+g(u)=f(x)$ in a domain $\text{Gw}\subset {\text{BBR}}^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$andthat$g$isnondecreasing,$g(0)\geq 0$and$g$ satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classical $C_{1,2}$ Wiener criterion then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition we discuss the question of uniqueness of large solutions.