Maximal solutions for $-\Delta u+u^q=0$ in open or finely open sets

TL;DR

This paper investigates the existence and uniqueness of solutions to the superlinear PDE $- riangle u + u^q=0$ in open and finely open sets, extending previous results to new classes of solutions with boundary blow-up conditions.

Contribution

It introduces new classes of solutions for the PDE in finely open sets and generalizes existing results on boundary blow-up solutions.

Findings

Established existence conditions for solutions in finely open sets.

Proved uniqueness of solutions under certain boundary conditions.

Extended classical results to more general set topologies.

Abstract

We study the existence and uniqueness of new classes of solutions of the superlinear equation $-\Delta u+u^q=0$ (q>1) in a domain of R^N or in a finely open set for the topology associated to the Bessel capacity C_{2,q'}. Condition of existence or uniqueness of solutions with boundary blow-up are obtained generalizing the results of Dhersin-Le Gall and of Labutin.