A perturbation theorem for the \(p\)-Laplace equation in unbounded domains
type: Item
instance of: scholarly article
title: A perturbation theorem for the \(p\)-Laplace equation in unbounded domains
zbMATH Open document ID: 1011.35052
DOI: 10.1016/S0362-546X(01)00762-3
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: This paper is devoted to the problem \[ \begin{cases} -\Delta_p u+ \lambda u^{p-1}= u^{q-1}\text{ in }\Omega\\ u>0\text{ in }\Omega,\;u\in W_0^{1, p} (\Omega),\end{cases} \tag{1} \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(\Delta_p\) is the \(p\)-Laplacian, \(\lambda>0\), \(1<p<q <p^* ={N p\over N-p}\), \(W^{1,p}_0(\Omega)\) is the well-known Sobolev space. It is well-known that, if \(p=2\) and \(\Omega'\) is an axial-symmetric unbounded domain and \(\Omega\) is the perturbation of \(\Omega'\), moreover \(\Omega'\) and \(\Omega\) have the same topology, then there exists no solution of (1) in \(\Omega'\), but in \(\Omega\) (1) has a solution. Here, the authors extend this result to the case \(p>1\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 1788013
zbMATH Keywords: Sobolev space
author: Xianling Fan
author: Feizhi Wang
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Equivalent Norms for Sobolev Spaces
cites work: A perturbation theorem for the equation −Δu + λu = uP in unbounded domains
cites work: Existence and non-existence results for semilinear elliptic problems in unbounded domains
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: The principle of symmetric criticality
cites work: Quasilinear elliptic boundary-value problems on unbounded cylinders and a related mountain-pass lemma
cites work: On harnack type inequalities and their application to quasilinear elliptic equations
cites work: Minimax theorems
full work available at URL: s0362-546x(01)00762-3
OpenAlex ID: W2039538307
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description: scientific article; zbMATH DE number 1788013