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.