A note on boundary blow-up problem of ∆u = u ^p

Assume that Ω is a bounded domain in R ⁿ with n ≥ 2. We study positive solutions to the problem, ∆u = u ^p in Ω, u(x) → ∞ as x → ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = u ^p . We show that a boundary blow-up solution exists in any bounded domain if 1 < p (Formula presented). In particular, when n = 2, there exists a boundary blow-up solution to ∆u = u ^p for all p ∈ (1, ∞). We also prove the uniqueness under the additional assumption that the domain satisfies the condition ∂Ω = ∂Ω.