Nontrivial solutions for some semilinear problems and applications to wave equations on balls and spheres
type: Item
instance of: scholarly article
title: Nontrivial solutions for some semilinear problems and applications to wave equations on balls and spheres
zbMATH Open document ID: 0863.47043
DOI: 10.12775/TMNA.1995.010
author: Yanyan Li
published in: Topological Methods in Nonlinear Analysis
publication date: 1997-06-03T00:00:00Z
review text: The abstract nonlinear equation \(Au=N(u)\) is studied in this paper. Here, \(A\) is a closed linear operator in a Hilbert space \(H\) satisfying some additional conditions and \(N\) is the (nonlinear) Nemitskii operator associated to a Carathéodory function \(g(x,s)\). The main assumption is that one simple eigenvalue of \(A\) crosses the function \(g(x,s)/s\) when \(s\) goes from \(-\infty\) to \(+\infty\). It is proved by using a continuation argument due to Berkovits and Mustonen that the above equation has at least two nontrivial solutions. This allows linear operators \(A\) which are not selfadjoint. Applications are given to both some semilinear wave equations on balls and spherical wave equations. Results depend on the space dimension \(n\), and the cases \(n=1,3\) are rather special.
reviewed by: Jesús Hernández
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 807920
zbMATH Keywords: continuation
zbMATH Keywords: periodic solution
zbMATH Keywords: multiple solutions
zbMATH Keywords: abstract nonlinear equation
zbMATH Keywords: closed linear operator
zbMATH Keywords: Nemitskii operator
zbMATH Keywords: Carathéodory function
zbMATH Keywords: semilinear wave equations on balls
zbMATH Keywords: spherical wave equations
MaRDI profile type: Publication
full work available at URL: tmna.1995.010
OpenAlex ID: W2571820491
description: scientific article; zbMATH DE number 807920