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

Abstract

We look for ground states and bound states $E:{\mathbb{R}}^3\to {\mathbb{R}}^3$ to the curl-curl problem $$\nabla \times (\nabla \times E)=f(x,E)in{\mathbb{R}}^3$$ 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 $\underset{s\to 0}{\lim }\Phi (s)/s^6=\underset{s\to +\infty }{\lim }\Phi (s)/s^6=0$. 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 Schrödinger equations.

to appear in Archive for Rational Mechanics and Analysis