Nontrivial solutions for superquadratic nonautonomous periodic systems
type: Item
instance of: scholarly article
title: Nontrivial solutions for superquadratic nonautonomous periodic systems
zbMATH Open document ID: 1209.34047
DOI: 10.12775/TMNA.2009.045
author: Shouchuan Hu
author: Nikolaos S. Papageorgiou
published in: Topological Methods in Nonlinear Analysis
publication date: 2010-07-09T00:00:00Z
review text: The authors consider a nonautonomous second order periodic system in \({\mathbb R}^n\) of the form \[ -x''(t)-A(t)x(t)=\nabla F(t,x(t)), \] where \(A:[0,b] \to {\mathbb R}^{n \times n}\) is continuous and such that \(A(t)\) is a symmetric matrix. The potential \(F\) satisfies the Carathéodory conditions and has superquadratic growth near infinity and near zero. It has to be remarked that the set of assumptions in this paper enables to treat also problems where the Ambrosetti-Rabinowitz condition may not be satisfied. The main result guarantees the existence of at least one nontrivial solution satisfying \(x(0)=x(b), \, x'(0)=x'(b)\). For the proof, a slight variant of a linking theorem due to \textit{S. Li} and \textit{M. Willem} [J. Math. Anal. Appl. 189, No.~1, 6--32 (1995; Zbl 0820.58012)] is applied.
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5734958
zbMATH Keywords: periodic solution
zbMATH Keywords: superquadratic potential
zbMATH Keywords: spectral resolution
zbMATH Keywords: local linking
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5734958