Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

Shaohong Wang,Zhan Zhou

doi:10.3390/math9141691

Abstract

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

Highlights

Throughout this article, we denote by R and Z the sets of real numbers and integers, respectively
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Mathematical models concerned with partial difference equations play important f,q roles in many fields

Summary

Introduction

Throughout this article, we denote by R and Z the sets of real numbers and integers, respectively. − um − r um−1 − 2um + um+1 = f um , (m, n) ∈ Z × N, where f is a real function defined on R These applications have greatly promoted the theoretical study of partial difference equations. (i, j) ∈ Z(1, m) × Z(1, n), Following the ideas from [34], the authors first investigated the nonlinear algebraic system f associated with (Eλ ) and further obtained several different results on the existence and f multiplicity of solutions for problem (Eλ ) by means of critical point theory. When the nonlinearity f satisfies appropriate hypotheses, we respectively acquire the existence of at least two or three positive solutions f ,q for problem (Dλ ) by applying the established strong maximum principle.

Preliminaries

Main Results

Examples

Conclusions