Semiclassical states for the curl–curl problem

Abstract

We show the existence of the so-called semiclassical states $U:R^3\to R^3$ to the following curl–curl problem ${\varepsilon }^2\nabla \times (\nabla \times U)+V\left(x\right)U=g\left(U\right),$ for sufficiently small $\varepsilon >0$. We study the asymptotic behaviour of solutions as $\varepsilon \to 0^{+}$ and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations.