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The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption
Author(s) -
Marcus Moshe,
Véron Laurent
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3037
Subject(s) - mathematics , sobolev space , boundary value problem , boundary (topology) , trace (psycholinguistics) , mathematical analysis , measure (data warehouse) , monotone polygon , domain (mathematical analysis) , degenerate energy levels , exponent , space (punctuation) , borel measure , pure mathematics , geometry , probability measure , physics , quantum mechanics , philosophy , linguistics , database , computer science
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.