On the first boundary value problem for nonlinear degenerate elliptic equations

This article is devoted to a proof of a general theorem on the existence of a solution of the first boundary value problem for a degenerate Bellman equation. In contrast to other papers the nonlinearity of the equation is used here and leads, for example, to a proof of solvability of the simplest Monge-Ampére equation for, in a strictly convex region of class with zero data on the boundary.