On the positive solutions of certain semi-linear elliptic equations (Q1009503)

This is a note on the class of semi-linear elliptic equations \[ \Delta u + f(x,u) + g(|x|) x \cdot \nabla u = 0, \] in exterior domains of Euclidean space of dimension \(n \geq 3\). Via theory for linear ordinary differential equations new sufficient conditions for the existence of positive solutions vanishing at infinity are given. In the case when the nonlinearity \(f(x,\cdot)\) has sublinear growth, and the radially symmetric function \(g\) is nonnegative, the authors show existence for a class of equations not covered by the criteria given by the reviewer [Nonlinear Anal., Theory Methods Appl. 64, No.~7 (A), 1608--1620 (2006; Zbl 1101.34022)]. Also other developments of those results are available in the literature.