Semiclassical stationary states for nonlinear Schr\"odinger equations under a strong external magnetic field

We construct solutions to the nonlinear magnetic Schr"odinger equation $$ \left{ \begin{aligned}

\varepsilon^2 \Delta_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & & $in$ \ \Omega,\ u &= 0 & & $on$ \ \partial\Omega, \end{aligned} \right. $$ in the semiclassical r'egime with strong magnetic fields. In contrast with the well-studied mild magnetic field r'egime, 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 the magnetic field. The 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 in the spirit of M. del Pino and P. Felmer.

We construct solutions to the nonlinear magnetic Schr"odinger equation $$ \left{ \begin{aligned}

\varepsilon^2 \Delta_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & & $in$ \ \Omega,\ u &= 0 & & $on$ \ \partial\Omega, \end{aligned} \right. $$ in the semiclassical r'egime with strong magnetic fields. In contrast with the well-studied mild magnetic field r'egime, 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 the magnetic field. The 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 in the spirit of M. del Pino and P. Felmer.

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