Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations Delta u = b(x)f(u)(C-0 + | backward difference u|(q)), x is an element of omega, where omega is a bounded domain with a smooth boundary in Double-struck capital R-N, C-0 > 0, q is an element of [0, 2), b is an element of Cloc alpha(omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \in C_{\rm{loc}}<^>\alpha(\Omega)$$\end{document} is positive in omega, but may be vanishing or appropriately singular on the boundary, f is an element of C[0,infinity), f(0) = 0, and f is strictly increasing on [0, infinity) (or f is an element of C(Double-struck capital R), f(s) > for all s is an element of Double-struck capital R, f is strictly increasing on Double-struck capital R). We show unified boundary behavior of such solutions to the problem under a new structure condition on f.