NLS ground state ($Q$)

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

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.

Normalize a minimizing sequence so that its $L^2$ norm and gradient norm are fixed. Symmetric decreasing rearrangement does not increase the gradient norm and preserves every $L^p$ norm, while radial compactness prevents translation loss. A weakly convergent subsequence therefore converges strongly in the nonlinear $L^{2+4/d}$ norm and yields a nonnegative minimizer.

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.

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 $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

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

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.