## Differential and Integral Equations

- Differential Integral Equations
- Volume 28, Number 11/12 (2015), 1197-1232.

### On quasi-variational inequalities with discontinuous multivalued lower order terms given by bifunctions

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 28, Number 11/12 (2015), 1197-1232.

**Dates**

First available in Project Euclid: 18 August 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1439901047

**Mathematical Reviews number (MathSciNet)**

MR3385140

**Zentralblatt MATH identifier**

1363.58011

**Subjects**

Primary: 58E35: Variational inequalities (global problems) 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40] 47J25: Iterative procedures [See also 65J15] 35J87: Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities [See also 35R35, 49J40]

#### Citation

Le, Vy Khoi. On quasi-variational inequalities with discontinuous multivalued lower order terms given by bifunctions. Differential Integral Equations 28 (2015), no. 11/12, 1197--1232. https://projecteuclid.org/euclid.die/1439901047