Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

Di Cosmo, Jonathan [UCL] Van Schaftingen, Jean [UCL]

We construct solutions to the nonlinear magnetic Schrödinger equation −ε²ΔA/ε2u+Vu=|u|p−2u in Ω, u=0 on ∂Ω, in the semiclassical régime under strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and on the magnetic field. Our results cover unbounded domains, fast-decaying electric potential and unbounded electromagnetic fields. The construction is variational and is based on an asymptotic analysis of solutions to a penalized problem following the strategy of M. del Pino and P. Felmer.