Positive solutions of a class of semilinear equations with absorption and Schrödinger equations

Abstract

Several results about positive solutions – in a Lipschitz domain – of a nonlinear elliptic equation in a general form Δu(x)−g(x,u(x))=0 are proved, extending thus some known facts in the case of g(x,t)=tq, 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 Schrödinger equations. A general result describing the measures that are diffuse with respect to certain capacities is also established and used. Appendix A by the first author provides classes of functions g such that the nonnegative solutions of Δu−g(.,u)=0 have some “good” properties that appear in the paper.