The Gagliardo-Nirenberg equation on Riemannian manifolds

Abstract

We establish geometric and analytic rigidity results for the Gagliardo-Nirenberg equation. On a complete noncompact Riemannian manifold with nonnegative Ricci curvature, the existence of a positive critical point of the Gagliardo-Nirenberg quotient forces the manifold to be Euclidean and the solution to be extremal. No minimizing assumption or volume growth hypothesis is required, thereby strengthening earlier rigidity results. In Euclidean space, we prove that all positive solutions of the normalized equation whose nonlinear mass on expanding annuli has sublinear growth are extremals. The linear threshold is optimal, as shown by explicit profiles that are constant in one direction. We also prove a sharp pointwise estimate involving the solution and its gradient, valid without assumptions at infinity, with equality at one point characterizing the extremals. The proofs use weighted divergence identities and a reflection argument controlled by annular mass. Explicit constructions reveal the geometric and variational limits of rigidity: nonflat quotient critical points exist beyond the curvature hypothesis, while the normalized equation admits positive solutions even under strictly positive sectional curvature and maximal volume growth.