Liouville theorems for a family of very degenerate elliptic non linear operators

Abstract

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\mathcal{P}}_k^{\pm }$, defined respectively as the sum of the largest and the smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\mathcal{P}}_k^{+}$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\mathcal{P}}_k^{-}$, we get totally different results.

arXiv comment15 pages