Asymptotic behavior of positive solutions of semilinear elliptic equations in $R^n$

Abstract

We will investigate the asymptotic behavior of positive solutions of the elliptic equation Δu+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0 {in} R^{n}. We establish that for $n\ge 3$ and $q>p>1$, any positive radial solution of (0.1) has the following property: ${\lim }_{r\to \infty }{r}^{\frac{2+l_1}{p-1}}u$ and ${\lim }_{r\to 0}{r}^{\frac{2+l_2}{q-1}}u$ always exist if $\frac{n+l_1}{n-2}<p<q,p\ne \frac{n+2+2l_1}{n-2},q\ne \frac{n+2+2l_2}{n-2}.$ In addition, we prove that the singular solution of (0.1) is unique under a certain condition

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