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

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem{−Δu=λu−b(x)h(u)in Ω,u=+∞on ∂Ω, where Ω is a smooth bounded domain in RN. 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)∼Hup 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)=Hup. 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 satisfyinglimd(x)→0u(x)KH−β(b∗(‖x−x0‖))−β=1, where d(x)=dist(x,∂Ω), b∗(r) and K are defined byb∗(r)=∫rR∫sRb(t)dtds,K=[β((β+1)C0−1)]1p−1,β:=1p−1. In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved.