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Pohozaev identity

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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

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2 / 2 / 4 / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 3 / 3 / 5 / Solution

Pohozaev identity for the mass-critical NLS ground state

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Pohozaev identity for the mass-critical NLS ground state

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