Decaying entire positive solutions of quasilinear elliptic equations
type: Item
instance of: scholarly article
title: Decaying entire positive solutions of quasilinear elliptic equations
zbMATH Open document ID: 0586.35043
DOI: 10.1007/BF01326845
author: Charles A. Swanson
author: Takaši Kusano
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178262
review text: In this paper the quasilinear elliptic equation \[ (*)\quad \Delta u+f(x,u,\nabla u)=0,\quad x\in {\mathbb{R}}^ N,\quad N\geq 3, \] is considered, where \(\nabla =(\partial /\partial x_ 1,...,\partial /\partial x_ N)\), \(\Delta =\nabla \cdot \nabla\), and f is nonnegative and locally Hölder continuous in \({\mathbb{R}}^ N\times {\mathbb{R}}_+\times {\mathbb{R}}^ N\), \({\mathbb{R}}_+=(0,\infty).\) Under additional structure hypotheses it is shown that equation (*) has an entire positive solution which decays to zero at infinity. In particular, conditions are established for the existence of an entire positive solution of (*) which behaves like a constant multiple of \(| x|^{2-N}\) as \(| x| \to \infty\). Examples to which our existence theory is applicable are \[ \Delta u+p(x)u^{\gamma}+\sum^{m}_{i=1}q_ i(x) u^{\rho_ i} | \nabla u|^{\sigma_ i}=0,\quad \Delta u+p(x)u^{- \gamma}+\sum^{m}_{i=1}q_ i(x) u^{-\rho_ i} | \nabla u|^{\sigma_ i}=0, \] where \(\gamma\), \(\rho_ i\), \(\sigma_ i\) are nonnegative constants satisfying \(0\leq \gamma <1\), \(0<\rho_ i+\sigma_ i<1\), \(i=1,...,m\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 3939838
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: Hölder continuous
zbMATH Keywords: positive solution
zbMATH Keywords: existence
zbMATH Keywords: existence theory
MaRDI profile type: Publication
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OpenAlex ID: W2056987311
description: scientific article; zbMATH DE number 3939838