Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4612678 | Journal of Differential Equations | 2009 | 20 Pages |
Abstract
In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem{âÎu=λuâb(x)h(u)in Ω,u=+âon âΩ, 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,â) and h(u)â¼Hup 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)â0u(x)KHâβ(bâ(âxâx0â))âβ=1, where d(x)=dist(x,âΩ), bâ(r) and K are defined bybâ(r)=â«rRâ«sRb(t)dtds,K=[β((β+1)C0â1)]1pâ1,β:=1pâ1. In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Zhifu Xie,