In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla u||_2^3$, where the constant $C$ is depending on $Q$.
Now if our domain is $\Omega=\mathbb{R}^N\setminus B_1(0)$, i.e., the exterior domain outside the unit ball, then we ask the question, what's the corresponding best constant? I.e., find the best $C_1$ s.t. $\|u\|_{4,\Omega}^4 \le $ $C_1$ $\|u\|_{2,\Omega}$
$\|\nabla u\|_{2,\Omega}^3$.
