Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 37, pp. 1-18.
Title: Multiple semiclassical states for singular magnetic
nonlinear Schrodinger equations
Author: Sara Barile (Politecnico di Bari, Italy)
Abstract:
By means of a finite-dimensional reduction, we show
a multiplicity result of semiclassical solutions
$u: \mathbb{R}^N \to\mathbb{C}$
to the singular nonlinear Schrodinger equation
$$
\Big( \frac{\varepsilon}{i} \nabla - A(x)\Big)^2 u +
u+(V(x)-\gamma(\varepsilon)W(x)) u = K(x) | u|^{p-1} u, \quad x \in
\mathbb{R}^N,
$$
where $N \geq 2$, $1 < p < 2^{*}-1$, $A(x), V(x)$ and $K(x)$
are bounded potentials. Such solutions concentrate near
(non-degenerate) local extrema or a
(non-degenerate) manifold of stationary points of an
auxiliary function $\Lambda$ related to the unperturbed electric
field $V(x)$ and the coefficient $K(x)$ of the nonlinear term.
Submitted November 26, 2007. Published March 14, 2008.
Math Subject Classifications: 35J10, 35J60, 35J20, 35Q55, 58E05.
Key Words: Nonlinear Schrodinger equations; external magnetic field;
singular potentials; semiclassical limit.