Existence of Solutions for the Discrete Dirichlet Problem Involving p-Mean Curvature Operator

Jianxia Wang,Zhan Zhou

doi:10.1155/2020/6591523

Abstract

This work is to discuss the Dirichlet boundary value problem of the difference equation with p -mean curvature operator. Under some determinate growth conditions on the nonlinear term, the existence of one solution or two nontrivial solutions is obtained via variational methods and some analysis techniques. Examples are also given to illustrate our theorems in the last section.

Highlights

Let N, Z, and R stand for the sets of natural numbers, integer numbers, and real numbers, respectively
Where b is a given positive integer, f(j, ·) ∈ C(R, R) for each j ∈ Z[1, b], Δ is the forward difference operator defined by Δu(j) u(j + 1) − u(j), and φp,c(t) ≔ (1 + |t|2)(p− 2)/2t, p ∈ (0, +∞)
It is easy to see that A has b distinct positive eigenvalues which are given by λj sin2􏼠 jπ 2(b +

Summary

Introduction

Let N, Z, and R stand for the sets of natural numbers, integer numbers, and real numbers, respectively. Where b is a given positive integer, f(j, ·) ∈ C(R, R) for each j ∈ Z[1, b], Δ is the forward difference operator defined by Δu(j) u(j + 1) − u(j), and φp,c(t) ≔ (1 + |t|2)(p− 2)/2t, p ∈ (0, +∞). Φp,c(Δu(j − 1)) is counted as a discretization of p -mean curvature operator. In order to research discrete boundary problems, different methods have been used: fixed point theorems, upper and lower solutions techniques, see [3, 5] and the references given therein. Compared to problem (4), problem (1) is much more difficult to handle because of p -mean curvature operator.

Mathematical Background

Main Results and Their Proofs

Examples