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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Louis Jeanjean; Stefan Le Coz
Journal: Trans. Amer. Math. Soc. 361 (2009), 5401-5416.
MSC (2000): Primary 35Q53, 35B35, 35A15, 35Q51
Posted: May 11, 2009
MathSciNet review: 2515816
Retrieve article in: PDF

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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 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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