Compactness of a mass-critical minimizing sequence
= Compactness of a mass-critical minimizing sequence
Suppose $\|u_n\|_2=\|Q\|_2$, $\|\nabla u_n\|_2=\|\nabla Q\|_2$, and $E(u_n)\to0$. The <profile decomposition modulo translations> and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence some translations $u_n(\cdot+x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.