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 $u:{\mathbb{R}}^3\to {\mathbb{R}}^3$ which yields a solution to the $p$-curl-curl equation in the critical case $$\nabla \times (|\nabla \times u{|}^{p-2}\nabla \times u)=|u{|}^{p^{\ast }-2}uin{\mathbb{R}}^3.$$ The cases $p=2$ and $p=3/2$ are motivated, respectively, by nonlinear Maxwell equations and zero modes of three-dimensional Dirac operators. The infinite-dimensional kernel of curl and the critical exponent prevent a direct application of standard compactness arguments. Our proof combines local compactness under the nonlinear constraint $\mathrm{div}(|u{|}^{p^{\ast }-2}u)=0$ and a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also establish existence and compactness results in axially symmetric classes, including two types of solutions for $p=3/2$ distinct from the explicit Loss--Yau fields. Finally, the same minimization strategy gives a new proof of compactness modulo translations and dilations for minimizing sequences in the classical critical Sobolev inequality.