{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 $\inf J=\frac{d}{d+2}\|Q\|_2^{4/d}$.


Pohozaev identity for the mass-critical NLS ground state

{c}
{wiki}

A Pohozaev identity is an integral identity for a nonlinear partial differential equation, obtained by multiplying the equation by a scaling vector field such as $x\mathbin{\cdot}\nabla u$ and applying <integration by parts>. It relates kinetic, potential, and nonlinear energies.


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 Incoming links: Pohozaev identity