Reduced measures on the boundary

We study the existence of solutions of the nonlinear problem  − u + g(u) =0 in , u = on , (0.1) where is a bounded measure and g : R → R is a nondecreasing continuous function with g(t) = 0, ∀t 0. Problem (0.1) admits a solution for every ∈ L1( ), but this neednot be the case when is a general bounded measure. We introduce a concept of reduced measure ∗ (in the spirit of Brezis et al. (Ann. Math. Stud., to appear)); this is the “closest” measure to for which (0.1) admits a solution. © 2004 Elsevier Inc. All rights reserved