Pohozaev identity for the mass-critical NLS ground state
= Pohozaev identity for the mass-critical NLS ground state
{c}
Testing the ground-state equation against $Q$ and applying the <Pohozaev identity> gives
$$
\|\nabla Q\|_2^2=\frac d2\|Q\|_2^2,
\qquad
\|Q\|_{2+4/d}^{2+4/d}=\frac{d+2}{d}\|\nabla Q\|_2^2.
$$
Consequently $E(Q)=0$ and $\inf J=\frac{d}{d+2}\|Q\|_2^{4/d}$.