Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

Ge Dong,Xiaochun Fang

doi:10.1155/2015/321437

Abstract

We consider the existence of solutions of variational inequality form. Findu∈D(J):〈A(u),v-u〉+〈F(u),v-u〉+J(v)-J(u)≥0,∀v∈W1LM(Ω),whose principal part is having a growth not necessarily of polynomial type, whereAis a second-order elliptic operator of Leray-Lions type,Fis a multivalued lower order term, andJis a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

Highlights

Let Ω be a bounded domain in RN (N ≥ 1) with Lipschitz boundary, and let A(u) = − div A(x, Du) be a Leray-Lions operator defined on W1,p(Ω), p ∈ (1, +∞)
It is well known that, in the study of differential equations, different classes of differential equations correspond to different function space settings
The classical Sobolev space is a special case of Orlicz-Sobolev spaces

Summary

Introduction

Let Ω be an open and bounded subset of RN and M an N-function. The Orlicz class KM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real valued measurable functions u on Ω such that ρM (u) < +∞ The closure in LM(Ω) of the set of bounded measurable functions with compact support in Ω is denoted by EM(Ω). The operator T : X → 2X∗ is called pseudomonotone if the following conditions hold.

Results

Conclusion