Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

Jianxia Wang,Zhan Zhou

doi:10.3390/math8030381

Abstract

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

Highlights

−4 φ p, c (4u(k − 1)) = λ f (k, u(k)), u(0) = u( T + 1) = 0, k ∈ Z(1, T ), where T is a given positive integer, λ is a positive real parameter, 4 is the forward difference operator defined by 4u(k ) = u(k + 1) − u(k ), f (k, ·) : R → R is a continuous function for each k ∈ Z(1, T ) and p −2 φ p, c (s) := (1 + |s|2 ) 2 s, p ∈ [1, +∞)
U = 0, x ∈ ∂Ω, where div φ p, c (Ou) is named p-mean curvature operator, which is a generalization of mean curvature operator; see [1,2]
If p = 1, it reduces to the mean curvature operator

Summary

Introduction

Consider the following Dirichlet boundary value problem of the nonlinear difference equation ( λ, f (Dp ). −4 φ p, c (4u(k − 1)) = λ f (k, u(k)), u(0) = u( T + 1) = 0, k ∈ Z(1, T ), where T is a given positive integer, λ is a positive real parameter, 4 is the forward difference operator defined by 4u(k ) = u(k + 1) − u(k ), f (k, ·) : R → R is a continuous function for each k ∈ Z(1, T ) and p −2 φ p, c (s) := (1 + |s|2 ) 2 s, p ∈ [1, +∞). 4 φ p, c (4u(k − 1)) may be seen as a discretization of the p-mean curvature operator. If p = 1, it reduces to the mean curvature operator. Several authors have discussed the existence and multiplicity of solutions of Problem (1); see [1,6,7,8,9,10,11,12]

Objectives

Conclusion