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 Pk−(D2u) or Pk+(D2u), some sort of “truncated Laplacians”, 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 P−k, we explicitly find the critical exponent p of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero Dirichlet boundary condition.