On a class of nonlocal nonlinear elliptic problems

Abstract

We give a direct approach to the solvability of a class of nonlocal problems which admit a formulation in terms of quasi- variational inequalities. We are motivated by nonlinear elliptic boundary value problems in which certain coefficients depend, in a rather general way, on the solution itself through global quantities like the total mass, the total flux or the total energy. We illustrate the existence results with several applications, including an implicit Signorini problem for steady diffusion of biological populations and a class of operator equations in nonlinear mechanics. We also discuss the non- uniqueness of the solutions.