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On Variational Inequalities with Multivalued Perturbing Terms Depending on Gradients

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Abstract

In this paper, we study variational inequalities of the form

$$\begin{aligned} \left\{ \begin{array}{l} \langle {\mathcal {A}}(u), v-u \rangle + \langle {{\mathcal {F}}}(u), v-u \rangle + J(v) - J(u) \ge 0,\quad \forall v\in X \\ u\in X, \end{array} \right. \end{aligned}$$

where \({\mathcal {A}}\) and \({{\mathcal {F}}}\) are multivalued operators represented by integrals, J is a convex functional, and X is a Sobolev space of variable exponent. The principal term \({\mathcal {A}}\) is a multivalued operator of Leray–Lions type. We concentrate on the case where \({{\mathcal {F}}}\) is given by a multivalued function \(f = f(x,u,\nabla u)\) that depends also on the gradient \(\nabla u\) 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.

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The author would like to thank the referees for their invaluable comments and suggestions.

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  1. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Vy Khoi Le

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Le, V.K. On Variational Inequalities with Multivalued Perturbing Terms Depending on Gradients. Differ Equ Dyn Syst 28, 763–790 (2020). https://doi.org/10.1007/s12591-017-0345-y

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