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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments


Authors: Louis Jeanjean and Stefan Le Coz
Journal: Trans. Amer. Math. Soc. 361 (2009), 5401-5416
MSC (2000): Primary 35Q53, 35B35, 35A15, 35Q51
Published electronically: May 11, 2009
MathSciNet review: 2515816
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Abstract: We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional action on a wide variety of constraints. For a general nonlinearity, we extend to the dimension $ N=2$ the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension $ N\geqslant3$.


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Additional Information

Louis Jeanjean
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: louis.jeanjean@univ-fcomte.fr

Stefan Le Coz
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France
Address at time of publication: Department of Mathematics, Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, 34014 Trieste, Italy
Email: slecoz@univ-fcomte.fr, lecoz@sissa.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04790-4
PII: S 0002-9947(09)04790-4
Received by editor(s): October 16, 2007
Published electronically: May 11, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.