Year: 2025
Type: preprint
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 $$ \nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\quad \hbox{in }\mathbb{R}^3.$$ 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 $\operatorname{div}(|u|^{p^*-2}u)=0$ 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.
Source: arXiv (Cornell University)
Authors: Mederski, Jarosław, Szulkin, Andrzej
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416437210
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