Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity

Authors:

Affiliation:

(1) Department of Mathematics, Henan Normal University, Xinxiang, 453002, Peoples Republic of China;(2) Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Abstract:

We consider the following semilinear elliptic equation with singular nonlinearity:

$Delta u - frac{1}{ u^alpha} + h(x) =0 quad hbox{in}, Omega$

where

and Ω is an open subset in ${mathbb R}^n, ngeq 2$ . Let u be a non-negative finite energy stationary solution and $Sigma=Big{ x in Omega: ; lim_{r to 0^+}{1}/{|B_r (x)|} int_{B_r (x)} |u| hbox{exists, and is equal to}, 0Big}$ be the rupture set of u. We show that the Hausdorff dimension of Σ is less than or equal to [(n−2) α+(n+2)]/(α +1).

Keywords:

KeywordHeading" >Mathematics Subject Classification Primary 35B45 Secondary 35J40

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