The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents
type: Item
instance of: scholarly article
title: The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents
zbMATH Open document ID: 0829.35041
author: Zhi-Qiang Wang
published in: Differential and Integral Equations
publication date: 1995-07-13T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^N\) \((N \geq 3)\) be a bounded domain with smooth boundary and let \(p = (N + 2)/(N - 2)\). The author is concerned with the existence and multiplicity of solutions \(u\) to the Neumann problem \[ - \Delta u + \lambda u = |u |^{p - 1} u \text{ in } \Omega, \quad \partial u/ \partial \nu = 0 \text{ on } \partial \Omega, \leqno (I)_\lambda \] where \(\lambda\) is a positive constant and \(\nu\) is the unit outer normal to \(\partial \Omega\). Denote also by \(H(x)\) the mean curvature of \(\partial \Omega\) at \(x\) with respect to \(\nu\). The aim of the paper is to establish connections between the topology of \(\partial \Omega\) and the number of nonconstant positive solutions \(u\) to \((I)_\lambda\). For instance, the author shows that, if \(H(x) > 0\) on \(\partial \Omega\) and \(\lambda\) is sufficiently large, then problem \((I)_\lambda\) possesses at least \(\text{cat} (\partial \Omega)\) nonconstant positive solutions. Actually, this is a corollary of a more general result, still involving the topology of \(\partial \Omega\).
reviewed by: Marco Degiovanni
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 763481
zbMATH Keywords: critical Neumann problem
zbMATH Keywords: multiplicity of solutions
zbMATH Keywords: mean curvature
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 763481