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November/December 2015 On quasi-variational inequalities with discontinuous multivalued lower order terms given by bifunctions
Vy Khoi Le
Differential Integral Equations 28(11/12): 1197-1232 (November/December 2015).

Abstract

In this paper, we consider the existence and some qualitative properties of solutions to quasi-variational inequalities of the form $$ \begin{cases} \langle {\mathcal A}(u) , v-u \rangle + \langle F(u,u) , v-u\rangle + J(v,u) - J(u,u) \ge 0,\;\forall v\in W^{1,p}(\Omega) , \\ u\in D(J(\cdot, u)) , \end{cases} $$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, ${\mathcal A}$ is a second-order elliptic operator of Leray--Lions type on $W^{1,p}(\Omega)$, $F(\cdot , \cdot)$ is a multivalued bifunction, and $J(\cdot , u)$ is a convex functional depending on $u$. We propose concepts of sub- and supersolutions for this problem and study the existence and enclosure of solutions, and also the existence of extremal solutions between its sub- and supersolutions. Properties and examples of the involved mappings $F$ and $J$ and of sub-supersolutions of the above quasi-variational inequality are also presented.

Citation

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Vy Khoi Le. "On quasi-variational inequalities with discontinuous multivalued lower order terms given by bifunctions." Differential Integral Equations 28 (11/12) 1197 - 1232, November/December 2015.

Information

Published: November/December 2015
First available in Project Euclid: 18 August 2015

zbMATH: 1363.58011
MathSciNet: MR3385140

Subjects:
Primary: 35J87, 47J20, 47J25, 58E35

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.28 • No. 11/12 • November/December 2015
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