Multiple semiclassical states for singular magnetic nonlinear Schrodinger equations

Abstract

By means of a finite-dimensional reduction, we show a multiplicity result of semiclassical solutions $u: mathbb{R}^N omathbb{C}$ to the singular nonlinear Schrodinger equation $$ Big( frac{varepsilon}{i} abla - A(x)Big)^2 u + u+(V(x)-gamma(varepsilon)W(x)) u = K(x) | u|^{p-1} u, quad x in mathbb{R}^N, $$ 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.

Keywords

• Nonlinear Schrodinger equations
• external magnetic field
• singular potentials
• semiclassical limit