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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.

The sharp inequality impliesThus energy controls the gradient whenever the mass is strictly below the mass of the NLS ground state.

The NLS ground state is the positive radial solution ofIt is, up to phase, translation, and scaling, the optimizer of the Sharp Gagliardo-Nirenberg inequalityConsequentlyEquality in the sharp inequality occurs precisely for the symmetry orbit of $Q$.

In dimension $d$, the mass-critical sharp inequality isand equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

For $p=2+4/d$, the mass-critical Weinstein functional isIt is invariant under multiplication by a nonzero scalar, spatial dilation, translation, and constant phase. Its positive minimizers are rescalings and translations of the NLS ground state.