Iterative Solutions for the Differential Equation with  p -Laplacian on Infinite Interval

Guoying Zhao,Ying Wang

doi:10.1155/2021/8765932

Abstract

This paper systematically investigates a class of fourth-order differential equation with p -Laplacian on infinite interval in Banach space. By means of the monotone iterative technique, we establish not only the existence of positive solutions but also iterative schemes under the suitable conditions. At last, we give an example to demonstrate the application of the main result.

Highlights

Introduction0, ð1Þ which is often used to describe, for example, diffusion process [1], with a spatial symmetric potential b, can be reduced to ðrðtÞφpðy′ðtÞÞÞ′ + cðtÞφpðyðtÞÞ = 0, where φpðxÞ = jxjp−2x, p > 1
The partial differential equation with the p-Laplacian operator −div À j∇u ðxÞjp−2 Á ∇uðxÞ + bðxÞφðuðxÞÞ =ð1Þ which is often used to describe, for example, diffusion process [1], with a spatial symmetric potential b, can be reduced to ðrðtÞφpðy′ðtÞÞÞ′ + cðtÞφpðyðtÞÞ = 0, where φpðxÞ = jxjp−2x, p > 1
It is frequent that only positive solutions are useful

Summary

Introduction

0, ð1Þ which is often used to describe, for example, diffusion process [1], with a spatial symmetric potential b, can be reduced to ðrðtÞφpðy′ðtÞÞÞ′ + cðtÞφpðyðtÞÞ = 0, where φpðxÞ = jxjp−2x, p > 1 This fact leads us to study the following p-Laplacian boundary value problem (BVP):. Many mathematical problems in science and engineering are set in unbounded domains, such as unsteady flow of gas through a semiinfinite porous media, the theory of drain flows, plasma physics, in determining the electrical potential in an isolated neutral atom In all these applications, it is frequent that only positive solutions are useful. To the best knowledge of the authors, there are few works in the literature dealing with the existence of positive solutions to boundary value problems of differential equation on infinite intervals with p-Laplacian operator by using iterative technique up to now. The goal of the present paper is to fill the gap in this area, so it is interesting and important to study the existence of positive solutions for BVP (2)

Preliminaries and Lemmas

Main Results

Example