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

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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).

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Authors and Affiliations

  1. Department of Mathematics, Henan Normal University, Xinxiang, 453002, Peoples Republic of China

    Zongming Guo & Juncheng Wei

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  1. Zongming Guo
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  2. Juncheng Wei
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Correspondence to Zongming Guo.

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Guo, Z., Wei, J. Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity. manuscripta math. 120, 193–209 (2006). https://doi.org/10.1007/s00229-006-0001-2

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  • DOI: https://doi.org/10.1007/s00229-006-0001-2

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