Positive solutions of some elliptic differential equations with oscillating nonlinearity

TL;DR

This paper investigates the existence of positive solutions that decay to zero at infinity for a class of elliptic differential equations with oscillating nonlinearities, using comparison methods to extend previous results.

Contribution

It introduces new conditions for positive solutions of elliptic equations with oscillating nonlinearities, complementing recent research in the field.

Findings

Established existence of solutions with decay at infinity

Extended previous results to broader classes of nonlinearities

Used comparison methods to analyze oscillating nonlinearities

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 $n\geq 3$, $g$ is nonnegative valued and $f$ has alternating sign, by means of the comparison method. Our results complement several recent contributions from [M. Ehrnstr\"{o}m, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154].