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

TL;DR

This paper reviews recent advances in the analysis of nonlinear curl-curl problems in three-dimensional space, focusing on ground and bound states related to nonlinear Maxwell equations using variational methods.

Contribution

It introduces a variational approach involving the generalized Nehari manifold to study nonlinear curl-curl problems with superlinear and critical nonlinearities.

Findings

Existence of ground and bound states established.

Refinements of previous results on nonlinear Maxwell problems.

Application of variational methods to strongly indefinite functionals.

Abstract

We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \quad\hbox{ in } \mathbb{R}^3,$$ 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.