Positive solutions of a class of semilinear equations with absorption and schr\"odinger equations

TL;DR

This paper investigates positive solutions of nonlinear elliptic equations with absorption in Lipschitz domains, extending known results for specific cases and combining PDE techniques with potential theory to characterize solutions and their boundary behavior.

Contribution

It generalizes existing results by characterizing positive solutions, boundary traces, and removability properties for a broad class of nonlinear elliptic equations using potential theoretic methods.

Findings

Characterization of removability via capacity

Description of boundary trace of solutions

Relation between positive and moderate solutions

Abstract

Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ \Delta u(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth domain. Our results include a characterization -in terms of a natural capacity- of a (conditional) removability property, a characterization of moderate solutions and of their boundary trace and a property relating arbitrary positive solutions to moderate solutions. The proofs combine techniques of non-linear p.d.e.\ with potential theoretic methods with respect to linear Schr\"odinger equations. A general result describing the measures that are diffuse with respect to certain capacities is also established and used. The appendix by the first author provides classes of functions $g$ such that the nonnegative solutions of $ \Delta u-g(.,u)=0$ has some "good" properties which appear in the paper.