On a class of essentially nonlinear elliptic differential–difference equations

An essentially nonlinear differential-difference equation containing the product of the $p$-Laplacian and a difference operator is considered. Sufficient conditions are obtained for the corresponding nonlinear differential-difference operator to be coercive and pseudomonotone in the case of nonvariational statement of the differential equation. The existence of a generalized solution to the Dirichlet problem for the nonlinear equation is proved.