A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent

Let $\Omega\subset \mathbb{R}^3$ be a Lipschitz domain and let $S_\mathrm{curl}(\Omega)$ the largest constant such that $$ \int_{\mathbb{R}^3}|\nabla\times u|^2\, dx\geq S_{\mathrm{curl}}(\Omega) \inf_{\substack{w\in W_0^6(\mathrm{curl};\mathbb{R}^3)\\ \nabla\times w=0}}\Big(\int_{\mathbb{R}^3}|u+w|^6\,dx\Big)^{\frac13} for any $u$ in $W_0^6(\mathrm{curl};\Omega)\subset W_0^6(\mathrm{curl};\mathbb{R}^3)$ where $W_0^6(\mathrm{curl};\Omega)$ is closure of $\mathcal{C}_0^{\infty}(\Omega,\mathbb{R}^3)$ $\{u\in L^6(\Omega,\mathbb{R}^3): u\in L^2(\Omega,\mathbb{R}^3)\}$ with respect to norm $(|u|_6^2+|\nabla\times u|_2^2)^{1/2}$. We show $S_{\mathrm{curl}}(\Omega)$ strictly larger than classical Sobolev $S$ $\mathbb{R}^3$. Moreover, independent $\Omega$ attained by ground state solution curl-curl problem (\nabla\times u) = |u|^4u if $\Omega=\mathbb{R}^3$. With aid those results, we also investigate states Brezis-Nirenberg-type operator bounded $$\nabla\times +\lambda u |u|^4u\quad\hbox{in }\Omega$$ so-called metallic boundary condition $\nu\times u=0$ on $\partial\Omega$, $\nu$ exterior normal $\partial\Omega$.