Pohozaev identity

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 \cdot \nabla u

and applying integration by parts. It relates kinetic, potential, and nonlinear energies.

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- Pohozaev identity for the mass-critical NLS ground state Pohozaev identity

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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Pohozaev identity for the mass-critical NLS ground state

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