Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with ϕc-Laplacian

Yanshan Chen,Zhan Zhou

doi:10.3390/sym12111839

Abstract

In this paper, based on critical point theory, we mainly focus on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator. Without imposing the symmetry or oscillating behavior at infinity on the nonlinear term f, we respectively obtain the sufficient conditions for the existence of at least three non-trivial solutions and the existence of at least two non-trivial solutions under different assumptions on f. In addition, by using the maximum principle, we also deduce the existence of at least three positive solutions from our conclusion. As far as we know, our results are supplements to some well-known ones.

Highlights

Let Z and R denote all integers and real numbers, respectively
In 2003, Guo and Yu in [8] used critical point theory for the first time to obtain sufficient conditions for the existence of periodic solutions and subharmonic solutions of difference equations. This crucial breakthrough inspired many scholars to use critical point theory to study the dynamics of difference equations and many meaningful and interesting results have been obtained, especially in periodic solutions [9,10,11], homoclinic solutions [12,13,14,15,16] and boundary value problems [17,18,19,20,21,22,23]
A discrete Dirichlet boundary value problem involving the mean curvature operator is studied in this paper

Summary

Introduction

Let Z and R denote all integers and real numbers, respectively. Let N be a fixed positive integer. Symmetry 2020, 12, 1839 sufficient conditions for the existence of at least three solutions of the following Dirichlet boundary value problem with φ p -Laplacian. Different from the conclusion of [18], Bonanno in [26] obtained the existence of three positive solutions without the asymptotic condition of the nonlinear function f. In [27], by using critical point theory, Nastasi and Vetro obtained the existence of at least two positive solutions to the following Dirichlet boundary value problem with ( p, q)-Laplacian. Let X be a real Banach space and let Φ, Ψ : X → R be two continuously Gâteaux differentiable functionals such that inf Φ = Φ(0) = Ψ(0) = 0. Φ(u)≤r points uλ, , uλ,2 such that Iλ (uλ,1 ) < 0 < Iλ (uλ,2 )

Preliminaries

Main Results

Examples

Conclusions