A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a Lipschitz domain and let $S_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}(\mathrm{\Omega })$ be the largest constant such that $${\int }_{{\mathbb{R}}^3}\mathrm{\mid }\mathrm{\nabla }\times u{\mathrm{\mid }}^{2}dx\ge S_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}(\mathrm{\Omega })\underset{\begin{array}{c}w\in W_0^6(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};{\mathbb{R}}^3)\\ \mathrm{\nabla }\times w=0\end{array}}{\inf }({\int }_{{\mathbb{R}}^3}\mathrm{\mid }u+w{\mathrm{\mid }}^{6}dx{)}^{\frac{1}{3}}$$ for any $u$ in $W_0^6(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};\mathrm{\Omega })\subset W_0^6(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};{\mathbb{R}}^3)$ where $W_0^6(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};\mathrm{\Omega })$ is the closure of ${\mathcal{C}}_0^{\mathrm{\infty }}(\mathrm{\Omega },{\mathbb{R}}^3)$ in ${u\in L^6(\Omega,$\mathbb{R}$^3): \nabla\times u\in L^2(\Omega,$\mathbb{R}$^3)}$ with respect to the norm $(\mathrm{\mid }u{\mathrm{\mid }}_6^2+\mathrm{\mid }\mathrm{\nabla }\times u{\mathrm{\mid }}_2^2{)}^{1\mathrm{/}2}$. We show that $S_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}(\mathrm{\Omega })$ is strictly larger than the classical Sobolev constant $S$ in ${\mathbb{R}}^3$. Moreover, $S_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}(\mathrm{\Omega })$ is independent of $\mathrm{\Omega }$ and is attained by a ground state solution to the curl-curl problem $$\mathrm{\nabla }\times (\mathrm{\nabla }\times u)=\mathrm{\mid }u{\mathrm{\mid }}^{4}u$$ if $\mathrm{\Omega }={\mathbb{R}}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $\mathrm{\Omega }$ $$\mathrm{\nabla }\times (\mathrm{\nabla }\times u)+\lambda u=\mathrm{\mid }u{\mathrm{\mid }}^{4}u\text{in }\mathrm{\Omega }$$ with the so-called metallic boundary condition $\nu \times u=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where $\nu$ is the exterior normal to $\mathrm{\partial }\mathrm{\Omega }$.