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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 $Q$.

By the Blowup alternative for the nonlinear Schrödinger equation, choose $t_n\uparrow T$ with $\Vert \nabla u(t_n){\Vert }_2\to \infty$, and setThe mass-critical scaling gives

The sequence is bounded and radial in $H^1$, so part 4 yields, after extraction, weak $H^1$ and strong $L^4$ convergence to some $v$. The energy identity gives $\Vert v_n{\Vert }_4^4\to 2\Vert \nabla Q{\Vert }_2^2$. The Sharp Gagliardo-Nirenberg inequality and weak lower semicontinuity force equality in every limiting norm inequality. Hence $v_n\to v$ strongly in $H^1$, and the variational characterization identifiesfor some $\gamma \in \mathbb{R}$; radiality removes translations and the gradient normalization fixes the scale.