Multiple solutions of second-order ordinary differential equation via Morse theory
type: Item
instance of: scholarly article
title: Multiple solutions of second-order ordinary differential equation via Morse theory
zbMATH Open document ID: 1270.34093
author: Qiong Meng
author: X. H. Tang
published in: Communications on Pure and Applied Analysis
publication date: 2013-04-29T00:00:00Z
review text: The authors consider the second-order ordinary differential equation \[ -\ddot{x}=f(t,x(t)) \] subject to the periodic boundary conditions \[ x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0, \] where \(f\in C([0,2\pi]\times\mathbb{R},\mathbb{R})\), \(f(t,x)\) is continuously differentiable in \(x\) for a.e. \(t\in [0,2\pi]\) and satisfies \(f(t,0)\equiv 0\) for all \(t\in[0,2\pi]\). The operator \(K=(-\frac{d^{2}}{dt^{2}}+I)^{-1}\) plays an important role. By using Morse index, Leray-Schauder degree and the Morse index theorem of Lazer-Solimini type, they obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.
reviewed by: Mohsen Timoumi
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 6159042
zbMATH Keywords: periodic solutions
zbMATH Keywords: Morse index
zbMATH Keywords: Leray-Schauder degree
zbMATH Keywords: Lazer-Solimini
MaRDI profile type: Publication
full work available at URL: cpaa.2012.11.945
OpenAlex ID: W2316286487
DOI: 10.3934/CPAA.2012.11.945
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description: scientific article; zbMATH DE number 6159042