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

Jeanjean, Louis; Le Coz, Stefan

doi:10.1090/S0002-9947-09-04790-4

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments
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Trans. Amer. Math. Soc. 361 (2009), 5401-5416 Request permission

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 $N\geqslant 3$.

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