Let $B = B_1(0)$ the unit open ball and consider the number $$ S : = \inf_{u \in H^1_0(B), u \neq 0} \frac{||u||^2_{H^1_0(B)}}{|u|_{L^{2^*}(B)}^2}. $$ I'm reading a paper which says that we can always consider a minimizing sequence $\{u_n\} \subset H^1_0(B)$ of $S$ where each term is a concentration based on a function $u \in C^{\infty}_0(B)$. I believe the authors meant that $u_n(x) = u(n x)$ and $$ \frac{||u_n||_{H^1_0(B)}}{|u_n|_{L^{2^*}(B)}^2} \rightarrow S. $$ How to prove that this is really true?
Here is the part of the paper which generate my doubt:
That this is possible follows from following:
The numerator and denominator scale the same under rescaling of both the function and $\mathbb{R}^n$. In particular, the ratio using $$ u_\epsilon = \epsilon^{-1}u(\epsilon^{-1}u), $$ is independent of $\epsilon > 0$.
The ratio is known to be bounded below because it it bounded below for all functions $f \in C^\infty_0(B)$. This is known as the Sobolev inequality.
But it's even simpler than that. The function that minimizes the ratio over all functions on $\mathbb{R}^n$ with bounded $H^1(\mathbb{R}^n)$ norm is known and has a simple explicit formula. This is known as the sharp Sobolev inequality and was proved independently by Aubin and Talent. A really really beautiful proof using optimal mass transport was given by Cordero, Nazaret, and Villani. So you can actually write down an explicit sequence of functions that both concentrate at the origin and minimize the ratio.
Let me add that the concept of concentrated compactness originated from the work of Talent and Aubin and Aubin's use of the sharp Sobolev inequality in his work on the Yamabe problem. It also played a crucial role in the work of Taubes and Uhlenbeck on self-dual Yang-Mills connections. At around the same time, people, such as Nirenberg, Brezis, Pierre-Louis Lions developed the same approach for certain types of PDE's, I believe it was Pierre-Louis Lions= who called it concentrated compactness.