Optimizers in Sobolev-curl inequalities

Abstract

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by nonlinear Maxwell equations and by the occurrence of zero modes in three-dimensional Dirac equations. Moreover, we introduce a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also consider existence of minimizers under some symmetry assumptions. Finally, our approach offers a new proof of the compactness of minimizing sequences for the Sobolev inequalities in the critical case.