Existence of Viscosity Solutions of Second Order Fully Nonlinear Elliptic Equations

We consider the problem of existence for viscosity solutions of seoond order fully nonlinear elliptic partial differential equations F(D²u, Du, u, z) = 0. We prove existence results for viscosity solutions in W^{1,∞} under assumptions that function F satisfies the natural structure conditions. We do not assume F is convex.