# Multiple solutions to a nonlinear curl-curl problem in $\mathbb{R}^3$

## TL;DR

This paper establishes the existence of multiple solutions, including a ground state, for a nonlinear curl-curl problem in three-dimensional space, using a novel critical point theory suited for strongly indefinite functionals.

## Contribution

It introduces a new critical point framework for strongly indefinite problems, proving multiple solutions for a nonlinear Maxwell-related curl-curl equation in D.

## Key findings

- Existence of a ground state solution.
- Existence of infinitely many bound states.
- Improvement over previous results on curl-curl problems.

## Abstract

We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from 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 controlled by an $N$-function $\Phi:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}\Phi(s)/s^6=\lim_{s\to+\infty}\Phi(s)/s^6=0$. We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in $\mathbb{R}^3$ and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schr\"odinger equations.