General uniqueness results and blow-up rates for large solutions of elliptic equations

Abstract

Making use of the Karamata regular variation theory and the López-Gómez localization method, we establish the uniqueness and asymptotic behaviour near the boundary ∂Ω for the large solutions of the singular boundary-value problem

where Ω is a smooth bounded domain in ℝ^N. The weight function b(x) is a non-negative continuous function in the domain, which can vanish on the boundary ∂Ω at different rates according to the point x₀ ∊ ∂Ω. f(u) is locally Lipschitz continuous such that f(u)/u is increasing on (0, ∞) and f(u)/u^p = H(u) for sufficiently large u and p > 1, here H(u) is slowly varying at infinity. Our main result provides a sharp extension of a recent result of Xie with f satisfying limu→f(u)/u^p = H for some positive constants H > 0 and p > 1.