On the positive solutions to some quasilinear elliptic partial differential equations

2009/04/09 by Octavian G. Mustafa, Yong Zhou, Mustafa, Octavian G. +1
Mathematics · Physics and Astronomy · #35A35 #35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35A35 #msc:35B40

paper · pdf · doi:10.48550/arxiv.0904.1489

arxiv created 2009/04/09 · arxiv updated 2009/12/01

Abstract

We establish that the elliptic equation Δu+f(x,u)+g(| x|)x⋅ ∇ u=0, where x∈ℝⁿ, n≥3, and | x|>R>0, has a positive solution which decays to 0 as | x|→ +∞ under mild restrictions on the functions f,g. The main theorem extends and complements the conclusions of the recent paper [M. Ehrnström, 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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