Uniqueness of Extremizers and Convergence of the Euler-Lagrange Iteration for the Critical Hardy-Littlewood-Sobolev Inequality

Let n ≥ 3, 2 ≤ α< n, and 1 < p,r < ∞ satisfy p⁻¹+r⁻¹=1+α/n. We prove uniqueness, up to common translations and dilations, of the normalized nonnegative extremizing pairs for the corresponding Hardy-Littlewood-Sobolev inequality. More generally, all positive finite-energy solutions of the Euler-Lagrange system belong to a single symmetry orbit. The proof compares two solutions at different relative scales. Total positivity of order two identifies the second eigenspace of an associated compact self-adjoint operator and forces each nonzero difference to change sign exactly once. The opposite orientations obtained at the two ends of the scale parameter then yield uniqueness. The same method proves radial nondegeneracy modulo dilation. We also establish strong convergence of the normalized alternating Euler-Lagrange iteration from every nonnegative, radially nonincreasing initial datum to a fixed extremizing pair, without rescaling the iterates. Square-root mass variables reduce the exclusion of several concentration profiles to a static spectral estimate and an energy-increment argument. The analysis includes sublinear exponents and the logarithmic borderline decay regimes.