= Pohozaev identity
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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 of
$$
-\Delta Q+Q-|Q|^{p-1}Q=0,
$$
write $A=\int|\nabla Q|^2$, $B=\int|Q|^2$ and $P=\int|Q|^{p+1}$. The identity is
$$
\frac{d-2}{2}A+\frac d2B=\frac d{p+1}P.
$$
One 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.
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