Liouville theorems for a family of very degenerate elliptic nonlinear operators

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators Pk ±, defined respectively as the sum of the largest and the smallest k eigenvalues of the Hessian matrix. For the operator Pk + 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 Pk −, we get totally different results.