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

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 Schrodinger 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 Delta u - g(., u) = 0 have some "good" properties that appear in the paper. (C) 2015 Elsevier Masson SAS. All rights reserved.