Positive solutions of some elliptic differential equations with oscillating nonlinearity

Abstract

We discuss the occurrence of positive solutions which decay to 0 as $|x|\to +\infty$ to the differential equation $\Delta u+f(x,u)+g(|x|)x\cdot \nabla u=0$, $|x|>R>0$, $x\in {\mathbb{R}}^n$, where $n\ge 3$, $g$ is nonnegative valued and $f$ has alternating sign, by means of the comparison method. Our results complement several recent contributions from [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154].