Existence of a Weinstein-functional minimizer
= Existence of a Weinstein-functional minimizer
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.