Semiclassical states for the curl-curl problem

TL;DR

The paper addresses the existence and asymptotic behavior of semiclassical states for a curl-curl Maxwell system with nonlinear polarization in the small-permeability limit, cast as $\nabla\times(\nabla\times \mathbf U) + V_\varepsilon(x)\mathbf U = g(\mathbf U)$ with $V_\varepsilon(x)=V(\varepsilon x)$ as $\varepsilon\to0^+$. It develops a symmetric variational framework on ${\mathcal G}(K)$-invariant spaces, proves continuous dependence of Nehari levels on the potential, analyzes the limiting Schrödinger-type problem with a singular term, and constructs semiclassical states via a Rabinowitz-type approach that concentrate according to the potential landscape. The main contributions include existence, $L^\infty$ bounds, decay rates, and precise localization (near $V_\infty$ or near minima of $V$) of curl-curl semiclassical states, together with a rigorous link between the curl-curl Maxwell problem and a singular Schrödinger problem. This work advances understanding of nonlinear Maxwell propagation in highly permeable media and clarifies localization phenomena in the semiclassical limit, providing a robust variational toolkit for similar curl-curl systems.

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 $$ \varepsilon^2\; \nabla \times (\nabla \times \mathbf{U}) + V(x) \mathbf{U} = g(\mathbf{U}), $$ 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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• Theorem 4.2