Onofri inequalities and rigidity results

Abstract

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.2010 Mathematics Subject Classification. Primary: 58J35, 58J05, 53C21; Secondary: 35J60, 35K55.

Rigidity results for the p‐Laplace type equations on compact Riemannian manifolds

“…Remark The result in Theorem 1.4 when $$$ n=2 $$$ reduces to the rigidity in [5]. …”

“…On the other hand, Dolbeault et al [5] continue to study the rigidity for a semilinear elliptic equation with exponential nonlinearity () on two‐dimensional compact Riemannian manifold $$$$ -\frac{1}{2}\Delta u+{\lambda}_{\tau }={e}^u. $$$$ …”

“…The advantage of this approach is that the value of 𝜗 can be directly calculated and does not require the assumption of positive Ricci curvature. On the other hand, Dolbeault et al [5] continue to study the rigidity for a semilinear elliptic equation with exponential nonlinearity (1.3)…”