On the positive solutions to some quasilinear elliptic partial differential equations

TL;DR

This paper proves the existence of positive solutions to a class of quasilinear elliptic PDEs in unbounded domains, extending previous results by analyzing the asymptotic behavior of solutions using comparison methods.

Contribution

It extends existing results on positive solutions of nonlinear elliptic equations by establishing decay properties under mild conditions on the functions involved.

Findings

Existence of positive solutions decaying to zero at infinity.

Extension of previous theorems to broader classes of equations.

Application of comparison methods and asymptotic analysis.

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 $| x|>R>0$, has a positive solution which decays to 0 as $| x|\to +\infty$ under mild restrictions on the functions $f,g$. The main theorem extends and complements the conclusions of the recent paper [M. Ehrnstr\"{o}m, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154]. Its proof relies on a general result about the long-time behavior of the logarithmic derivatives of solutions for a class of nonlinear ordinary differential equations and on the comparison method.