Rigidity results for semilinear elliptic equations with exponential nonlinearities and Moser-Trudinger-Onofri inequalities on two-dimensional Riemannian manifolds

Jean Dolbeault (1) , Maria J. Esteban (1) , Gaspard Jankowiak (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 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

Mots clés: Onofri's inequality; Moser-Trudinger inequality; Sobolev spaces; Sobolev inequality; Riemanian manifolds; extremal functions; best constants; fast diffusion equation; rigidity; carré du champ; uniqueness

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Rigidity results for semilinear elliptic equations with exponential nonlinearities and Moser-Trudinger-Onofri inequalities on two-dimensional Riemannian manifolds. 2014. ⟨hal-00985211v1⟩