Optimizers in Sobolev-curl inequalities

TL;DR

This paper investigates a Sobolev inequality involving the p-curl operator in three dimensions, establishing existence of minimizers and solutions to related equations, and introduces a novel variational method for strongly indefinite problems.

Contribution

It introduces a new variational approach for quasilinear strongly indefinite problems and proves existence of minimizers in Sobolev inequalities involving the p-curl operator.

Findings

Existence of minimizers for the p-curl Sobolev inequality.

Solutions to the p-curl-curl equation in the critical case.

A new proof of compactness of minimizing sequences.

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.