Global continuum and multiple positive solutions to one-dimensional p-Laplacian boundary value problem

Liping Wei,Ruyun Ma

doi:10.1186/s13662-020-02667-0

Abstract

We show the global structure of the set of positive solutions of a discrete Dirichlet problem involving the p-Laplacian difference operator suggesting suitable conditions on the weight function and nonlinearity. We obtain existence and multiplicity of positive solutions for λ lying in various intervals in mathbb {R} by using the directions of a bifurcation and the Picone-type identity for discrete p-Laplacian operators.

Highlights

Introduction and main result LetT > 1 be an integer, T := [1, T]Z = {1, 2, . . . , T}, T := {0, 1, . . . , T + 1}
We are concerned with existence and multiplicity of positive solutions of the discrete boundary value problem
Existence of positive solutions for discrete boundary value problems involving the pLaplacian difference operator has been studied by several authors, we refer to Agarwal et al [1], Chu and Jiang [4], and [7, 8, 10, 12] as well as the references therein

Summary

Introduction

We are concerned with existence and multiplicity of positive solutions of the discrete boundary value problem It is the purpose of this paper to show that (1.1) has three positive solutions for λ lying in various intervals in R suggesting suitable conditions on the weight function and nonlinearity by using the directions of a bifurcation and the Picone-type identity (for related results, we refer to [6, 19, 22]) for discrete p-Laplacian operators due to Řehák [20]. 2, we show the existence of bifurcation from the first eigenvalue for the corresponding problem according to the standard argument and the rightward direction of bifurcation.

Results

Conclusion