On positive solutions of fully nonlinear degenerate Lane-Emden type equations

We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either ${\mathcal{P}}_k^{-}(D^{2}u)$ or ${\mathcal{P}}_k^{+}(D^{2}u)$, some sort of \lq\lq truncated Laplacians\rq\rq, given respectively by the smallest and the largest partial sum of $k$ eigenvalues of the Hessian matrix. New phenomena with respect to the semilinear case occur. Moreover, for ${\mathcal{P}}_k^{-}$, we explicitely find the critical exponent $p$ of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero Dirichlet boundary condition.