Asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth
type: Item
instance of: scholarly article
title: Asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth
zbMATH Open document ID: 1036.34066
DOI: 10.1016/S0362-546X(02)00149-9
author: Zongming Guo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The author considers a perturbation of the equation \[ - \Delta_p u = u^k \quad {\text{in }} B,\qquad u = 0 \quad{\text{in }}\partial B, \] (\(B\) is the unit ball in \(\mathbb{R}^N\), \(N>2\), and \(\Delta_p\) the \(p\)-Laplacian) which has no positive solutions. The singularly perturbed equation is \[ - \Delta_p u = u^k - \varepsilon u^q \quad {\text{in }} B, \qquad u = 0\quad {\text{in }} \partial B, \tag \(*\) \] which has at least two positive radial solutions, one a small solution, the other a large solution. A small solution to \((*)\) is a function \(u_{\epsilon}\) such that \[ \limsup_{\varepsilon \to 0}{\varepsilon}^{1/(q-k)} \| u_{\varepsilon} \| _{\infty} = c, \quad c \in [0,1). \] The author studies the behaviour of the small solutions \(u_{\varepsilon}\) to \((*)\) as \(\varepsilon \to 0\) giving first an upper bound on \(u_{\varepsilon}\) (which implies that \(u_{\varepsilon}(x) \to 0\) for every \(x \in B \setminus \{0\}\)) and then giving the behaviour of \(\| u_{\varepsilon} \| _{\infty}\).
reviewed by: Fabio Paronetto
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 1879993
zbMATH Keywords: Asymptotic behaviour
zbMATH Keywords: small solutions
zbMATH Keywords: quasilinear elliptic equations
zbMATH Keywords: critical and supercritical growth
MaRDI profile type: Publication
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full work available at URL: s0362-546x(02)00149-9
OpenAlex ID: W1996815255
description: scientific article; zbMATH DE number 1879993