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

Louis Jeanjean (1) , Stefan Le Coz (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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>2.

Mots clés: Instability; standing waves; nonlinear Klein-Gordon equations; mountain-pass arguments; blow-up; variational methods

Domaines: Equations aux dérivées partielles [math.AP]

Louis Jeanjean, Stefan Le Coz. Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments. Transactions of the American Mathematical Society, 2009, pp.Trans. Amer. Math. Soc. 361 (2009), 5401-5416. ⟨10.1090/S0002-9947-09-04790-4⟩. ⟨hal-00183260⟩