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

TL;DR

This paper studies the asymptotic behavior of positive solutions to a class of semilinear elliptic equations in ^n, establishing existence of limits at zero and infinity for radial solutions and proving uniqueness of singular solutions under specific conditions.

Contribution

It provides new results on the asymptotic limits of positive radial solutions and the uniqueness of singular solutions for a class of elliptic equations with weighted nonlinearities.

Findings

Limits of solutions at zero and infinity exist under certain conditions.

Positive radial solutions exhibit specific asymptotic behaviors.

Singular solutions are unique under particular constraints.

Abstract

We will investigate the asymptotic behavior of positive solutions of the elliptic equation \Delta u+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0 {in} R^{n}. We establish that for $n\geq 3$ 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 $\lim_{r\to0}r^{\frac{2+l_{2}}{q-1}}u$ 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 solution of (0.1) is unique under a certain condition