Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

Scott Armstrong (1) , Boyan Sirakov (2)

1 Department of Mathematics
2 MODAL'X - Modélisation aléatoire de Paris X

We study fully nonlinear elliptic equations such as $F(D^2u) = u^p; p > 1;$ in $R^n$ or in exterior domains, where F is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of F, that sharply characterizes the range of p > 1 for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Veron as well as Cutri and Leoni, who found critical exponents for supersolutions in the whole space Rn, in case F is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.

Domaines: Equations aux dérivées partielles [math.AP]

Scott Armstrong, Boyan Sirakov. Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2011, 10, pp.711-728. ⟨hal-00450368v2⟩