We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.
Squassina, M., Secchi, S. (2005). On the location of spikes for the Schrödinger equation with electromagnetic field. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 7(2), 251-268 [10.1142/S0219199705001738].
On the location of spikes for the Schrödinger equation with electromagnetic field
SECCHI, SIMONE
2005
Abstract
We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.
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