 Pohozaev identity for the mass-critical NLS ground state

ID: pohozaev-identity-for-the-mass-critical-nls-ground-state

Testing the ground-state equation against

Q

and applying the Pohozaev identity gives

∥\nabla Q ∥_{2}^{2} = \frac{d}{2} ∥ Q ∥_{2}^{2}, ∥ Q ∥_{2 + 4/ d}^{2 + 4/ d} = \frac{d + 2}{d} ∥\nabla Q ∥_{2}^{2} .

Consequently

E (Q) = 0

and

in f J = \frac{d}{d + 2} ∥ Q ∥_{2}^{4/ d}

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