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

We survey recent results concerning ground states and bound states $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ to the curl-curl problem $$\mathrm{\nabla }\times (\mathrm{\nabla }\times u)+V(x)u=f(x,u)\text{ in }{\mathbb{R}}^3,$$ which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\mathrm{\nabla }\times (\mathrm{\nabla }\times \cdot )$. 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.