On positive solution to a second order elliptic equation with a singular nonlinearity

Galina V. Grishina

doi:10.3934/cpaa.2010.9.1335

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2010, Volume 9, Issue 5: 1335-1343. Doi: 10.3934/cpaa.2010.9.1335

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On positive solution to a second order elliptic equation with a singular nonlinearity

Galina V. Grishina^1,

1.
Department of Applied Mathematics, Bauman Moscow State Technical University, 2-aya Baumanskaya, 5, 105005 Moscow

Received: July 31, 2009

Revised: September 30, 2009

Published: August 2010

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Abstract

We consider a second order elliptic equation with measurable bounded coefficients

$ (a_{ij}(x)u_{x_i})_{x_j}+p(x)|x|^su^{-\sigma}=0, x\in\Omega \setminus \{ O\}, $

where $\sigma >0$, $s$ is any real number, and $\Omega\subset R^n$, $n\ge 3$ is a bounded domain, which contains the origin $O$.
The aim of this paper is to establish existence, nonexistence and behavior of positive weak solutions near the isolated singularity $O$.

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Mathematics Subject Classification: Primary: 35J15, 35J61, 35J75; Secondary: 35A01, 35Bxx.

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