Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. (English) Zbl 1062.81056
Summary: We obtain multiple solutions \(u:\mathbb{R}^N\to\mathbb{C}\) of the nonlinear Schrödinger equation with an external magnetic field,
\[
\left( \frac hi\nabla-A(x)\right)^2u+\bigl(U(x)-E\bigr)u=f(x,u),\quad x\in \mathbb{R}^N,
\]
where \(N\geq 2\), \(A\) is a real-valued vector magnetic potential, \(U\) is a real electric potential function and the nonlinear term \(f(x,t)\) grows subcritically in \(t\). The number of solutions to the equation is shown to be bounded below by some number which depends on the category of a set defined by some properties of \(V\) and the coefficients of the nonlinear term. We perform appropriate changes of gauges which are made on functions which are concentrated around points lying in some well-defined manifold.
MSC:
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
Keywords:
Nonlinear Schrödinger equations; External magnetic field; Semiclassical; standing waves; Variational methodsReferences:
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