On the positive solutions of certain semi-linear elliptic equations

Abstract

We establish that the elliptic equation $\Delta u+f(x,u)+g(|x|)x\cdot \nabla u=0$, where $x\in\mathbb{R}^{n}$, $n\geq3$, and $|x| >A>0$, has a positive solution which decays to $0$ as $|x|\rightarrow +\infty$ under mild restrictions on the functions $f,g$. The main theorem improves substantially upon the conclusions of the recent paper [M. Ehrnström, Positive solutions for second-order nonlinear differential equations, Nonlinear Anal. TMA 64 (2006), 1608--1620]. Its proof relies on a sharp result of non-oscillation of linear ordinary differential equations and on the comparison method.