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Some properties of positive radial solutions for some semilinear elliptic equations

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  • 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.

    Mathematics Subject Classification: 35B45; 35J40; 35J25.


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