Solution of Nonlinear Elliptic Boundary Value Problemsand Its Iterative Construction

Li Wei,Liling Duan,Haiyun Zhou

doi:10.1155/2012/210325

Abstract

We study a kind of nonlinear elliptic boundary value problems with generalized p‐Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

Highlights

The study on nonlinear boundary value problems with p-Laplacian operator, Δp, is a hot topic since it has a close relationship with practical problems
In Wei and Zhou 5, we established that 1.2 has solutions in Lp Ω, where 2 ≤ p < ∞, and in Wei 6 we proved that 1.2 has solutions in Ls Ω, where max N, 2 ≤ p ≤ s < ∞
As the summary and extension of 5, 6, we studied the following nonlinear boundary value problem:

Summary

Introduction

The study on nonlinear boundary value problems with p-Laplacian operator, Δp, is a hot topic since it has a close relationship with practical problems. From Lemma 3.2 by Wei and Agarwal 7 , we know that Bp,q is everywhere defined, monotone, hemicontinuous, and coercive. ∂Φp is maximal monotone in view of Lemma 2.3.

Results

Conclusion