Semiclassical states for the curl-curl problem

Abstract

We show the existence of the so-called semiclassical states $\mathbf{U}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ to the following curl-curl problem $${\epsilon }^2\nabla \times (\nabla \times \mathbf{U})+V(x)\mathbf{U}=g(\mathbf{U}),$$ for sufficiently small $\epsilon >0$. We study the asymptotic behaviour of solutions as $\epsilon \to 0^{+}$ 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.