On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents

We are concerned in this paper with variational inequalities of the form: { 〈 A ( u ) , v − u 〉 + 〈 F ( u ) , v − u 〉 ⩾ 〈 L , v − u 〉 , ∀ v ∈ K , u ∈ K , where A is a maximal monotone operator, F is an integral multivalued lower order term, and K is a closed, convex set in a Sobolev space of variable exponent. We study both coercive and noncoercive inequalities. In the noncoercive case, a sub-supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub- and supersolutions.