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

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.