Multiplicity of nodal solutions for elliptic equations with supercritical exponent in contractible domains
type: Item
instance of: scholarly article
title: Multiplicity of nodal solutions for elliptic equations with supercritical exponent in contractible domains
zbMATH Open document ID: 0901.35034
DOI: 10.12775/TMNA.1996.031
author: Donato Passaseo
published in: Topological Methods in Nonlinear Analysis
publication date: 1998-12-02T00:00:00Z
review text: Consider the problem \[ \Delta u+| u|^{p-2} u=0 \quad\text{in } \Omega,\;u=0 \quad\text{on } \partial \Omega,\;u\not \equiv 0\quad\text{in } \Omega \tag{P} \] in a bounded domain \(\Omega \subset \mathbb{R}^n\), \(n\geq 3\), with \(p>2n/(n-2)\). The main difficulty in this problem is the supercritical exponent in the nonlinearity, which causes a lack of compactness in the embedding \(H_0^{1,2} (\Omega) \hookrightarrow L^p (\Omega)\). Moreover, the shape of \(\Omega\) seems to play an important role. If \(\Omega\) is starshaped, no solution of (P) can exist due to the Pokhozhaev identity. There are also examples of domains, having a nontrivial topology in the sense of \textit{A. Bahri} and \textit{J. M. Coron} [Commun. Pure Appl. Math. 41, 253-294 (1988; Zbl 0649.35033)], such that (P) has no solution. In contrast there exist contractible domains with an arbitrarily large number of positive solutions. Here, for all \(h\in \mathbb{N}\) the author constructs a contractible domain \(\Omega_h\) such that (P) has at least \(h^2\) nodal \((u^+ \not \equiv 0,\;u^-\not \equiv 0)\) solutions.
reviewed by: Ralf Beyerstedt
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1127697
MaRDI profile type: Publication
full work available at URL: tmna.1996.031
OpenAlex ID: W2596630320
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description: scientific article; zbMATH DE number 1127697