Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian

Sijia Du,Zhan Zhou

doi:10.3390/math8112030

Abstract

Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results.

Highlights

Let Z, R, N denote all integers, real numbers and positive integers, respectively
∆2 x (i, j) = x (i, j + 1) − x (i, j), ∆21 x (i, j) = ∆1 (∆1 x (i, j)) and ∆22 x (i, j) = ∆2 (∆2 x (i, j)), φ p is the p-Laplacian operator given by φ p (s) = |s| p−2 s, 1 < p < +∞ and f ((i, j), ·) ∈ C (R, R) for all (i, j) ∈ Z(1, m) × Z(1, n)
The study of difference equations has captured special attention, which is due to the fact that difference equations are widely used as mathematical models in discrete optimization, physics, population genetics, etc. [1,2,3,4]

Summary

Introduction

Let Z, R, N denote all integers, real numbers and positive integers, respectively. Define Z( a, b) =. Until now, there is very little research on the partial difference equations with the p-Laplacian For this reason, this paper is to study the existence of multiple solutions for partial discrete Dirichlet problems involving the p-Laplacian. In the framework of variational f methods, we consider the two-dimensional discrete problem (Sλ ) by using critical point theory and we come up with more specific sets of parameters such that the existence of infinitely many solutions for f problem (Sλ ) can be obtained.

Preliminaries

Main Results

Examples

Conclusions