Pohozaev identity

A Pohozaev identity is an integral relation obtained by testing a nonlinear partial differential equation against a scaling direction such as $x\cdot \nabla Q$. For a sufficiently regular, rapidly decaying real solution ofwrite $A=\int |\nabla Q{|}^2$, $B=\int |Q{|}^2$ and $P=\int |Q{|}^{p+1}$. The identity isOne derivation uses the stationary energy functional $S(Q)=A/2+B/2-P/(p+1)$. Evaluate it on $Q(x/\lambda )$: the three terms scale as ${\lambda }^{d-2}$, ${\lambda }^d$ and ${\lambda }^d$. Differentiating at $\lambda =1$ gives the displayed formula, with decay justifying the scaling variation. Direct integration by parts with $x\cdot \nabla Q$ gives the same identity.

Testing against $Q$ also gives $A+B=P$. Combining the two relations at $p=1+4/d$ yields the Pohozaev identity for the mass-critical NLS ground state. The usefulness is that a differential equation imposes exact constraints on global kinetic, mass and nonlinear integrals.

Testing the ground-state equation against $Q$ and applying the Pohozaev identity givesConsequently $E(Q)=0$ and $\inf J=\frac{d}{d+2}\Vert Q{\Vert }_2^{4/d}$, where $J$ is the Weinstein functional.