On the positive solutions to some quasilinear elliptic partial differential equations

Details On the positive solutions to some quasilinear elliptic partial differential equations On the positive solutions to some quasilinear elliptic partial differential equations

2009-04-09

arxiv.org/abs/0904.1489v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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