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Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity

Published online by Cambridge University Press:  18 September 2007

Zongming Guo  and
Juncheng Wei
Zongming Guo
Affiliation:
Department of Mathematics, Henan Normal University, Xinxiang 453002, People's Republic of China and Dong Hua University, Shanghai 200051, People's Republic of China (guozm@public.xxptt.ha.cn)
Juncheng Wei
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong (wei@math.cuhk.edu.hk)
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Abstract

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We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity:

$$ \Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2, $$

where $\nu>0$. In order to apply the MMP, it is crucial to obtain the asymptotic expansion of $u$ at $\infty$.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 5 , October 2007 , pp. 963 - 994
Copyright
2007 Royal Society of Edinburgh