Multiple sign-changing solutions for a semilinear Neumann problem and the topology of the configuration space of the domain boundary
type: Item
instance of: scholarly article
title: Multiple sign-changing solutions for a semilinear Neumann problem and the topology of the configuration space of the domain boundary
zbMATH Open document ID: 1200.35076
author: Naoki Shioji
published in: Calculus of Variations and Partial Differential Equations
publication date: 2010-07-06T00:00:00Z
review text: In this paper, the main aim of the author is to establish a lower estimate for the number of sign-changing solutions to the following Neumann problem \[ -d^2 \Delta u + u =f(u)\quad \text{in }\Omega,\qquad\frac{\partial u}{\partial \nu}=0 \quad \text{in }\partial \Omega, \tag{P} \] where \(d > 0\) is small enough, \(\Omega \subset\mathbb R^N\) with \( (N \geq 2)\) is a bounded or unbounded domain whose boundary is nonempty, compact and smooth, and the nonlinearity \(f \in C(\mathbb R,\mathbb R)\) is Sobolev subcritical. Under some further conditions, the author proved that there exists \(d_0>0\) such that for each \(d\in (0, d_0)\), problem (P) has at least \(\text{cat}(C(\partial\Omega)\times [0,1]^2, C(\partial\Omega)\times [0,1]^2)\) sign-changing solutions, and each of them has precisely two nodal domains. Furthermore, if \(\Omega\) is bounded, then there is at least one other sign-changing solution which has at most four nodal domains.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5730930
zbMATH Keywords: second order elliptic equations
zbMATH Keywords: variational methods
zbMATH Keywords: sign-changing solution
zbMATH Keywords: Neumann boundary value problem
reviewed by: Huansong Zhou
MaRDI profile type: Publication
full work available at URL: s00526-009-0288-4
OpenAlex ID: W4240704564
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DOI: 10.1007/S00526-009-0288-4
description: scientific article; zbMATH DE number 5730930