Coupled systems of nonlinear variational inequalities and applications

In this paper we investigate the existence of solutions for a system consisting of two inequalities of variational type. Each inequality is formulated in terms of a nonlinear functional χ and ψ, respectively and a coupling functional B. We consider two sets of assumptions (Hχi), (Hψj) and (HBk), i,j,k∈{1,2} and we show that, if the constraints sets are bounded, then a solution exists regardless if we assumed the first or the second hypothesis on χ, ψ or B, thus obtaining eight possibilities. When the constraint sets are unbounded a coercivity condition is needed to ensure the existence of solutions. We provide two such conditions. An application, arising from Contact Mechanics, in the form of a partial differential inclusion driven by the Φ-Laplace operator is presented in the last section.