Multiple positive solutions for a class of Neumann problems

Hongliang Gao,Ruyun Ma

doi:10.14232/ejqtde.2015.1.48

Abstract

We study the existence of multiple positive solutions of the Neumann prob- lem u 00 (x) = lf(u(x)), x2 (0, 1), u 0 (0) = 0 = u 0 (1),

Highlights

In this paper, we are concerned with the existence of multiple positive solutions to the Neumann problem−u (x) = λ f (u(x)), x ∈ (0, 1), (1.1)u (0) = 0 = u (1), where λ is a positive parameter, f ∈ C([0, ∞), R) and for some β > 0 such that f (0) = 0, f (s) > 0 for s ∈ (β, +∞), f (s) < 0 for s ∈ (0, β), and θ (> β) is the unique positive zero of F(s) =The Neumann problems have played a significant role in mathematical physics, and have attracted the attention of many researchers over the last two decades, see [3, 9, 11] andH
We study the existence of multiple positive solutions of the Neumann problem
We are concerned with the existence of multiple positive solutions to the Neumann problem

Summary

Introduction

We are concerned with the existence of multiple positive solutions to the Neumann problem. The existence and multiplicity of positive solutions for the Neumann boundary value problems were investigated in connection with various configurations of f by the fixed point theorems in [3, 11] and by a detailed analysis of time-map associated with (1.1) in [9]. In [8], Maya and Shivaji obtained multiple positive solutions for a class of semilinear elliptic boundary value problems by using sub-super solutions arguments when f ∈ C1 satisfies the following conditions:. Maya and Shivaji [8] obtained multiple positive solutions for a class of semilinear elliptic boundary value problems when f satisfies (f1)–(f4).

Some notations and auxiliary results

Proof of the main result