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

Abstract

We construct solutions to the nonlinear magnetic Schrödinger equation $$\{\begin{array}{rlrl}-{\epsilon }^2{\Delta }_{A/{\epsilon }^2}u+Vu& =|u{|}^{p-2}u& & \text{in}\Omega ,\\ u& =0& & \text{on}\partial \Omega ,\end{array}$$ in the semiclassical régime with 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 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.

23 pages, minor corrections

References in corpus (2)

• Single--peaks for a magnetic Schrödinger equation with critic al growth
• Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities

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