Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem {Delta u +/- a(x)vertical bar del u vertical bar(q) = b(x)f(u), x is an element of Omega, u(x) = infinity, x is an element of partial derivative Omega. Here a(x) is an element of C-alpha(Omega) is a positive function with alpha is an element of (0, 1), b(x) is an element of C-alpha(Omega) is a non-negative function and may be singular near the boundary or vanish on the boundary, and the nonlinear term f is a Gamma-varying function, whose variation at infinity is not regular. We focus our investigation on the existence and asymptotic behavior close to the boundary partial derivative Omega of large solutions by Karamata regular variation theory and the method of upper and lower solution. The main results of this paper emphasize the central role played by the gradient term vertical bar del u vertical bar(q) and the weight functions a(x) and b(x). Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.