Nonlinear Schrödinger equation with bounded magnetic field. (English) Zbl 1442.35360
Summary: The paper studies existence of solutions for the nonlinear Schrödinger equation
\[
- (\nabla + i A ( x ) )^2 u + V(x) u = f(| u |) u
\]
with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is required. Solutions are obtained by means of a general structural statement about bounded sequences in the magnetic Sobolev space.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
| 46B50 | Compactness in Banach (or normed) spaces |
Keywords:
Schrödinger operator; magnetic field; ground state; concentration compactness; profile decomposition; critical pointsReferences:
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