Nondegeneracy and uniqueness for boundary blow-up elliptic problems

n this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Delta u = lambda a(x)u(P) + g(x, u) in Omega, with u = +infinity on partial derivative Omega, has a unique positive solution for large enough lambda, and determine its asymptotic behavior as lambda -> infinity. Here p > 1, a(x) is a continuous function which can be singular near partial derivative Omega and g(x, u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u(p) by e(u) or a "logistic type" function f (u). (c) 2005 Elsevier Inc. All rights reserved.