American Institute of Mathematical Sciences

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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