On Variational Inequalities with Multivalued Perturbing Terms Depending on Gradients

In this paper, we study variational inequalities of the form ⟨ A ( u ) , v - u ⟩ + ⟨ F ( u ) , v - u ⟩ + J ( v ) - J ( u ) ≥ 0 , ∀ v ∈ X u ∈ X , where A and F are multivalued operators represented by integrals, J is a convex functional, and X is a Sobolev space of variable exponent. The principal term A is a multivalued operator of Leray–Lions type. We concentrate on the case where F is given by a multivalued function f = f ( x , u , ∇ u ) that depends also on the gradient ∇ u of the unknown function. Existence of solutions in coercive and noncoercive cases are considered. In the noncoercive case, we follow a sub-supersolution approach and prove the existence of solutions of the above inequality under a multivalued Bernstein–Nagumo type condition.