Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations

Byeon, Jaeyoung; Oshita, Yoshihito

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 4, pp. 1121-1152.

Nous recollons des ondes stationnaires d'ordres différents en énergie, se concentrant autour de points critiques d'un potentiel V. Nous introduisons une méthode hybride, utilisant à la fois une méthode de réduction de Lyapunov–Schmidt, et une méthode variationnelle pour recoller des ondes stationnaires, se concentrant en des minima locaux, éventuellement sans équation-limite correspondante, et d'autres se concentrant en des points critiques quelconques, convergeant vers des solutions de problèmes-limites correspondants, satisfaisant une condition de non-dégénérescence.

We glue together standing wave solutions concentrating around critical points of the potential V with different energy scales. We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those concentrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition.

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