On nonlinear and quasiliniear elliptic functional differential equations

We consider nonlinear ellipticfunctional differential equations. The corresponding operator has the form of a product of nonlinear elliptic differential mapping and linear difference mapping. It were obtained sufficient conditions for solvability of the Dirichlet problem. A concrete example shows that a nonlinear differential--difference operator may not be strongly elliptic even if the nonlinear differential operator is strongly elliptic and the linear difference operator is positive definite. The analysis is based onthe theory of pseudomonotone--type operators and linear theory of ellipticfunctional differential operators.