The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

Source abstract

Abstract In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation −Δ u + g ( x, u ) = 0 in a smooth domain Ω ⊂ ℝ N , for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g α,q of the form g α,q ( x, u ) = d ( x , ∂Ω) α | u | q −1 u , with α > −2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities { g α,q }, determine the critical value of the exponent q (for fixed α > −2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for −Δ u + g α,q ( x, u ) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.