Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk
type: Item
instance of: scholarly article
title: Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk
zbMATH Open document ID: 1254.35089
author: Yasuhito Miyamoto
published in: Advances in Differential Equations
publication date: 2011-10-05T00:00:00Z
review text: Let \(D\) denote the open unit ball in \({\mathbb R}^2\), and assume that \(f:{\mathbb R}\rightarrow {\mathbb R}\) is a differentiable function such that \(f(0)=0\) and \(f'(0)=1\). This paper is concerned with the structure of sign-changing solutions of the nonlinear elliptic equation \(\Delta u+\lambda f(u)=0\) in \(D\) under the Dirichlet condition \(u=0\) on \(\partial D\). The main result of the present paper establishes the existence of infinitely many bifurcation points from which unbounded continua of nodal solutions emanate, whereby the eigenfunctions corresponding to each bifurcation point are nonradially symmetric. The proofs combine several tools from nonlinear analysis, including bifurcation theory, variational methods, Bessel functions and Morse theory.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B36
Mathematics Subject Classification ID: 35J91
zbMATH DE Number: 5953772
zbMATH Keywords: Morse index
zbMATH Keywords: variational methods
zbMATH Keywords: Bessel functions
zbMATH Keywords: infinitely many bifurcation points
zbMATH Keywords: nodal solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5953772