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

Authors:

Institution:

(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 $\alpha >1, h(x) \in C^1 (\Omega)$ and Ω is an open subset in ${\mathbb R}^n, n\geq 2$ . Let u be a non-negative finite energy stationary solution and $\Sigma=\Big\{ x \in \Omega: \lim_{r \to 0^+}{1}/{|B_r (x)|} \int_{B_r (x)} |u| \hbox{exists, and is equal to}\, 0\Big\}$ 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:

Mathematics Subject Classification" target="_blank">Mathematics Subject Classification Primary 35B45 Secondary 35J40

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