On Variational Inequalities with Multivalued Perturbing Terms Depending on Gradients
- PDF / 673,587 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 9 Downloads / 211 Views
On Variational Inequalities with Multivalued Perturbing Terms Depending on Gradients Vy Khoi Le1
© Foundation for Scientific Research and Technological Innovation 2017
Abstract 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. Keywords Variational inequality · Generalized pseudomonotone mapping · Multivalued mapping · Sobolev space with variable exponent · Bernstein–Nagumo condition Mathematics Subject Classification 58E35 · 47J20 · 47J25 · 35J87
Introduction We are concerned in this paper with variational inequalities of the form A(u), v − u + F (u), v − u + J (v) − J (u) ≥ 0, ∀v ∈ X u ∈ X,
B 1
(1.1)
Vy Khoi Le [email protected] Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
123
Differ Equ Dyn Syst
where A and F are multivalued operators given by integrals, J is a convex functional, and X is a Sobolev space of variable exponent. The principal term A, given (formally) by A(x, ∇u)∇vd x, (1.2) A(u), v =
is a multivalued operator of Leray–Lions type, F represents the lower order term, and J describes the constraints imposed on the problem. There is a rich literature on (ordinary) first and second order differential inclusions (cf. e.g. [4,5,15,29,30,34,50,51,59] and the extensive references therein). On the other hand, it was studied in [43] and [49] (partial) differential inclusions containing the p-Laplacian and multivalued nonlinearities; solvability of inclusions with multivalued mappings of monotone types was investigated in [2,3,24,31] by topological methods. Some sub-supersolutions methods applied to related variational equations and inequalities can be found, for example, in [7–11,33,38,40] and the references therein. We are interested here in the case where F is given by a multivalued function f that depends also on the gradient of the unknown function u, F (u), v = f (x, u, ∇u)vd x, ∀u, v ∈ X, (1.3)
to 2R . Furthermore, A and f have variable growth rates. where f The problem is thus formulated in an appropriate Sobolev space with variable exponents, and because of dependence of f on ∇u, our problem is intrinsically of nonvariational nature. We would like to study the existence and some properties of solutions of (1.1) in both coercive and noncoercive cases. As noted, variational methods are not available for our problem here. For the coercive case, we
Data Loading...
