A NOTE ON BOUNDARY BLOW-UP PROBLEM OF Δu = up

Assume that Omega is a bounded domain in R-n with n >= 2. We study positive solutions to the problem, Delta u = u(p) in Omega, u(x) -> infinity as x -> partial derivative Omega , where p > 1. Such solutions are called boundary blow-up solutions of Delta u = u(p). We show that a boundary blow-up solution exists in any bounded domain if 1 < p < n/n-2 . In particular, when n = 2, there exists a boundary blow-up solution to Delta u = u(p) for all p is an element of (1, infinity). We also prove the uniqueness under the additional assumption that the domain satisfies the condition partial derivative Omega = partial derivative(Omega) over bar.