Uniqueness and blow-up rate of large solutions for elliptic equation−Δu=λu−b(x)h(u)<

Abstract

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problemwhere Ω is a smooth bounded domain in R N . The weight function b(x) is a non-negative continuous function in the domain. h(u) is locally Lipschitz continuous and h(u)/u is increasing on (0, ∞) and h(u) ∼ Hu p for sufficiently large u with H > 0 and p > 1. Naturally, the blow-up rate of the problem equals its blow-up rate for the very special, but important, case when h(u) = Hu p . We distinguish two cases: (I) Ω is a ball domain and b is a radially symmetric function on the domain in Theorem 1.1; (II) Ω is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in Theorem 1.2. The blow-up rate is explicitly determined by functions b and h. In case (I), the singular boundary value problem has a unique solution u satisfying lim d(x)→0In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved.