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The Pohozaev identity for the mass-critical NLS ground state givesSince $Q$ is a minimizer,

Part 3 and Mass conservation for the nonlinear Schrödinger equation giveThe last relation and the energy formula implywhere the final equality follows from the Pohozaev identity for the mass-critical NLS ground state. Thus $(v_n)$ is a minimizing sequence for the Weinstein functional with the same normalization as $Q$.

Apply the profile decomposition from part 4. The Sharp Gagliardo-Nirenberg inequality bounds each profile by the product of its gradient energy and its mass to the power $2/d$. Since the total mass is exactly $\Vert Q{\Vert }_2^2$, any split into two nonzero profiles would make the limiting inequality strict. Hence precisely one profile carries all the mass and gradient energy. The norm decouplings then make the remainder converge strongly to zero in $H^1$. For suitable translations $x_n$,and the Sobolev embedding theorem gives the required strong convergence in $L^{2+4/d}$. This is the compactness of a mass-critical minimizing sequence.

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.