Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field. (English) Zbl 1315.35200
Summary: We construct solutions to the nonlinear magnetic Schrödinger equation
\[
\begin{cases} - \varepsilon^2 \Delta_{A / \varepsilon^2} u + V u = | u |^{p - 2} u & \text{ in } \Omega, \\ u = 0 & \text{ on } \partial \Omega, \end{cases}
\]
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. A. del Pino and P. L. Felmer [Calc. Var. Partial Differ. Equ. 4, No. 2, 121–137 (1996; Zbl 0844.35032); J. Funct. Anal. 149, No. 1, 245–265, Art. No. FU963085 (1997; Zbl 0887.35058); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, No. 2, 127–149 (1998; Zbl 0901.35023)].
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B25 | Singular perturbations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 78A25 | Electromagnetic theory (general) |
Keywords:
magnetic nonlinear Schrödinger equation; strong magnetic field; Lorentz force on a magnetic dipole; penalization method; asymptotics and concentration; estimates for distributional solutionsReferences:
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