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Spectral asymptotics for nonlinear Sturm-Liouville problems

Spectral asymptotics for nonlinear Sturm-Liouville problems Abstract. We consider nonlinear Sturm-Liouville problems with power nonlinearities. We shall establish an asymptotic formula of eigenvalues with respect to the L2 norm of corresponding eigenfunctions. Our result is deeply connected with the properties of ground state of the corresponding equation. 1980 Mathematics Subject Classification: 34B15, 34B25. 1. Introduction We consider the following nonlinear Sturm-Liouville problem m V } f~«"(*) = l"WrX*)\-(\ u(0) = «(1) = 0, 0<<1, where p > l and e R is a real parameter. The main object of this paper is to investigate the asymptotic behavior of with respect to «,= , where u,,tlt is the eigenfunction corresponding to , which has exactly n -- l interior zeroes in (0, 1) and is positive near = 0. It was shown in [1] that for (1), the following asymptotic formula holds for p = 3 by using elliptic functions: (2) «1 ael^J^. T. Shibata Our purpose of this paper is to establish an asymptotic formula such s (2) for general p > l and themain workconcerns the case n = l, since the proof of thecase n > 2 is obtained by the same methods s those used in the case n = 1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Spectral asymptotics for nonlinear Sturm-Liouville problems

Shibata, Tetsutaro
Forum Mathematicum , Volume 7 (7) – Jan 1, 1995
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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1995.7.207
Publisher site
See Article on Publisher Site

Abstract

Abstract. We consider nonlinear Sturm-Liouville problems with power nonlinearities. We shall establish an asymptotic formula of eigenvalues with respect to the L2 norm of corresponding eigenfunctions. Our result is deeply connected with the properties of ground state of the corresponding equation. 1980 Mathematics Subject Classification: 34B15, 34B25. 1. Introduction We consider the following nonlinear Sturm-Liouville problem m V } f~«"(*) = l"WrX*)\-(\ u(0) = «(1) = 0, 0<<1, where p > l and e R is a real parameter. The main object of this paper is to investigate the asymptotic behavior of with respect to «,= , where u,,tlt is the eigenfunction corresponding to , which has exactly n -- l interior zeroes in (0, 1) and is positive near = 0. It was shown in [1] that for (1), the following asymptotic formula holds for p = 3 by using elliptic functions: (2) «1 ael^J^. T. Shibata Our purpose of this paper is to establish an asymptotic formula such s (2) for general p > l and themain workconcerns the case n = l, since the proof of thecase n > 2 is obtained by the same methods s those used in the case n = 1.

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 1995

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