Sufficient and Necessary Conditions on the Existence and Estimates of Boundary Blow-Up Solutions for Singular p-Laplacian Equations

Let Ω denote a smooth, bounded domain in ℝN (N ≥ 2). Suppose that g is a nondecreasing C1 positive function and assume that b(x) is continuous and nonnegative in Ω, and that it may be singular on ∂Ω. In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta p}u = b(x)g(u)\,\,{\rm{for}}x\, \in \,\Omega ,\,\,u(x) \to + \infty \,{\rm{as}}\,{\rm{dist}}\,{\rm{(}}x,\partial \Omega) \to 0.\,$$\end{document} The estimates of such solutions are also investigated. Moreover, when b has strong singularity, the nonexistence of boundary blow-up (radial) solutions and infinitely many radial solutions are also considered.