Maximal Solutions of Semilinear Elliptic Equations with Locally Integrable Forcing Term

TL;DR

This paper investigates the existence and boundary behavior of maximal solutions to semilinear elliptic equations with locally integrable forcing, establishing conditions under which solutions blow up at the boundary and discussing their uniqueness.

Contribution

It introduces new conditions for the existence of large solutions in semilinear elliptic equations with locally integrable forcing terms, extending previous results.

Findings

Maximal solutions blow up on the boundary under Wiener criterion.

Existence of large solutions is established for certain boundary conditions.

Discussion on the uniqueness of large solutions is provided.

Abstract

We study the existence of a maximal solution of $-\Gd u+g(u)=f(x)$ in a domain $\Gw\subset \BBR^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$ and that $g$ is nondecreasing, $g(0)\geq 0$ 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.