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 $||u||_{L^4}^4\leq C||u||_{L^2}||\nabla u||_{L^2}^3$, where the constant $C$ is depending on $Q$.

Now if our domain is $\Omega=\mathbb{R}^N\B_1(0)$, 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 $C_1$ such that $||u||_{L^4(\Omega)}^4\leq C_1||u||_{L^2(\Omega)}||\nabla u||_{L^2(\Omega)}^3$.