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.

to 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