Nonlinear curl-curl problems in ${\mathbb{R}}^3$

Abstract

We survey recent results concerning ground states and bound states $u:{\mathbb{R}}^3\to {\mathbb{R}}^3$ to the curl-curl problem $$\nabla \times (\nabla \times u)+V(x)u=f(x,u)in{\mathbb{R}}^3,$$ which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla \times (\nabla \times \cdot )$. The growth of the nonlinearity $f$ is superlinear and subcritical at infinity or purely critical and we demonstrate a variational approach to the problem involving the generalized Nehari manifold. We also present some refinements of known results.

arXiv commentto appear in the special issue of Minimax Theory and its Applications: The Nehari manifold: theory, applications, related topics. arXiv admin note: substantial text overlap with arXiv:1901.05776

References in corpus (3)

• Perfect Optical Solitons: Spatial Kerr Solitons as Exact Solutions of Maxwell's Equations
• The Brezis-Nirenberg problem for the curl-curl operator
• Uncountably many solutions for nonlinear Helmholtz and curl-curl equations with general nonlinearities