July  2010, 9(4): 943-953. doi: 10.3934/cpaa.2010.9.943

Some properties of positive radial solutions for some semilinear elliptic equations

1. 

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received  June 2009 Revised  November 2009 Published  April 2010

We are interested in the singular elliptic equation

$ \Delta h = \frac{1}{\alpha } h^{ -\alpha }-p(r)$ in $R^N( N\geq3 ),$

where $ \alpha >1$ and the monotone decreasing function $p(r)$ satisfying $\lim_{r\rightarrow \infty}p(r)=c>0.$ In this paper we show that for any $ \eta >0 $ there is a unique radial solution $ h(r) $ with $ h(0)=\eta $ and $ h(r)$ is oscillatory in $ [0, \infty )$. We prove $ \lim_{r \rightarrow \infty } h(r)=( \alpha c)^{-\frac{1}{\alpha}}. $ We also obtain similar properties of singular solutions, of which the zero set is nonempty.

Citation: Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943
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