We establish the comparison principle, existence and regularity of solutions to the following problem concerning the mixed operator:where M + λ,Λ is the extremal Pucci's operator and (−∆ À N ) s denotes the fractional sub-Laplacian on Heisenberg group.

We consider a class of degenerate elliptic fully nonlinear equations with applications to Grad equations:where γ ≥ 1 is a constant, Ω is a bounded domain in R N with C 1,1 boundary. We prove the existence of a W 2,p -viscosity solution to the above equation, which degenerates when the gradient of the solution vanishes.