J. Korean Math. Soc. 2011; 48(2): 431-447
Printed March 1, 2011
https://doi.org/10.4134/JKMS.2011.48.2.431
Copyright © The Korean Mathematical Society.
Baishun Lai, Qing Luo, and Shuqing Zhou
Henan University, Henan University, Hunan Normal University
We investigate the asymptotic behavior of positive solutions to the elliptic equation \begin{equation} \Delta u+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0\ \mbox{in}\ \ \mathbb R^n.\ \end{equation} We obtain a conclusion that, for $n\geq 3, -2 < l_{2} < l_{1}\leq 0$ and $q > p >1$, any positive radial solution to (0.1) has the following properties: $\lim_{r\to\infty}r^{\frac{2+l_{1}}{p-1}}u$ and $\lim_{r\to0}r^{\frac{2+l_{2}}{q-1}}u$ always exist if $\frac{n+l_{1}}{n-2} < p < q, \ \ p\neq\frac{n+2+2l_{1}}{n-2},\ \ q \neq\frac{n+2+2l_{2}}{n-2}.$ In addition, we prove that the singular positive solution of (0.1) is unique under some conditions.
Keywords: semilinear elliptic equation, positive solutions, asymptotic behavior, singular solutions
MSC numbers: Primary 35J60; Secondary 35B05, 35B40
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