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

Armstrong, Scott N.¹ ; Sirakov, Boyan²

¹ Department of Mathematics       Louisiana State University Baton Rouge, LA 70803
² UFR SEGMI Université Paris 10 92001 Nanterre Cedex, France and CAMS, EHESS 54 bd Raspail 75270 Paris Cedex 06, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 729-746

Résumé

We study fully nonlinear elliptic equations such as

$$F\left(D^{2}u\right)=u^p,p>1,$$

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-Véron [6] as well as Cutri and Leoni [11], who found critical exponents for supersolutions in the whole space ${ℝ}^n$, 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.

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Publié le : 2018-06-21

Classification : 35B53, 35J60