Homotopical linking and Morse index estimates in min-max theorems
type: Item
instance of: scholarly article
title: Homotopical linking and Morse index estimates in min-max theorems
zbMATH Open document ID: 0834.58015
DOI: 10.1007/BF02570474
author: Miguel Ramos
author: Luis Sanchez
published in: Manuscripta Mathematica
publication date: 1996-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156087
review text: Let \(E\) be a real Hilbert space and \(f\in C^2(E, R)\). It is shown in this paper that if \(f\) satisfies certain conditions of saddle or linking type, then the associated minimax level must necessarily contain a critical point whose Morse index satisfies an explicitly given bound (upper or lower -- depending on the minimax procedure chosen). This extends earlier results by \textit{A. C. Lazer} and \textit{S. Solimini} [Nonlinear Anal., Theory Methods Appl. 12, No. 8, 761-775 (1988; Zbl 0667.47036)] and \textit{S. Solimini} [Manuscr. Math. 63, No. 4, 421-453 (1989; Zbl 0685.58010)]. Applications are given to the problem of obtaining multiple solutions for ordinary and elliptic partial differential equations of second-order. In particular, it is shown that under suitable conditions on \(g\) and whenever \(T\) is sufficiently large, the equation \(\ddot u(t)+ g(u(t))= 0\) has at least \(k\) nonconstant solutions of period \(kT\).
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 784074
zbMATH Keywords: Morse index
zbMATH Keywords: homotopical linking
zbMATH Keywords: multiple solutions
zbMATH Keywords: subharmonic solutions
MaRDI profile type: Publication
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OpenAlex ID: W1988969280
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description: scientific article; zbMATH DE number 784074