Weinstein functional ($J(u)$)

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.