Subharmonic solutions of nonlinear wave equations on \(S^n\)
type: Item
instance of: scholarly article
title: Subharmonic solutions of nonlinear wave equations on \(S^n\)
zbMATH Open document ID: 0990.35098
DOI: 10.1007/BF01783477
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2000-01-04T00:00:00Z
review text: The paper is concerned with the existence of \(2k\pi\)-periodic solutions of the wave equation (*)~\(u_{tt}-\Delta_{n}u = g(u,t)\), where \(\Delta_{n}\) is the Laplace-Beltrami operator on the sphere \(S^n\), \(g\) is \(2\pi\)-periodic in \(t\), superlinear in \(u\) as \(|u|\to\infty\) and \(g(u,0)\equiv 0\). Moreover, \(g_{u}(0,t)=\lambda\), where \(\lambda\) is not in the spectrum of \({\mathcal L} = \partial_{tt}-\Delta_{n}\) in \(L^2(S^n\times[0,2\pi])\). The authors show that if \(n\) is even, then for each large prime number \(k\) there exists a subharmonic solution \(u_{k}\) to (*) which has minimal period \(2k\pi\). Moreover, \(u_{k}\to 0\) in \(L^p(S^n\times[0,2k\pi])\) as \(k\to\infty\). If \(n\) is odd, the same results remain valid under more restrictive hypotheses on \(g\) (which are needed because in this case the operator \(\mathcal L\) has an eigenvalue of infinite multiplicity). The proof employs an abstract minimax principle of \textit{V. Benci, A. Capozzi} and \textit{D. Fortunato} [Ann. Mat. Pura Appl. (4) 143, 1-46 (1986; Zbl 0632.34036)]. That the solutions \(u_{k}\) have minimal period \(2k\pi\) and not \(2\pi\) follows from the fact that \(u_{k}\to 0\) in \(L^p\) and 0 is an isolated \(2\pi\)-periodic solution of (*). Some results on the existence of multiple subharmonics are also proved in this paper for \(g\) odd in \(u\) or independent of \(t\). The functional corresponding to the problem is then respectively \({\mathbb Z}_{2}\)- and \(S^1\)-invariant.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 1383225
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: minimax principle
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: periodic solution
zbMATH Keywords: eigenvalue of infinite multiplicity
author: Alberto Capozzi
author: Addolorata Salvatore
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1383225