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Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

Set $p=2+4/d$. For $u_{a,\lambda }(x)=au(\lambda x)$, the change of variables $y=\lambda x$ givesandThe factors cancel, so the Weinstein functional satisfies $J(u_{a,\lambda })=J(u)$.

The Gagliardo-Nirenberg interpolation inequality givesConsequently $J(u)\ge C_d^{-1}$ for every nonzero $u\in H^1$, and therefore $I>0$.

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.