Existence of multiple positive solutions for one-dimensional p-Laplacian

Abstract

In this paper we consider the multipoint boundary value problem for one-dimensional p-Laplacian φ p (u ) + f (t, u) = 0, t ∈ (0, 1), subject to the boundary value conditions:where φ p (s) = |s| p−2 s, p > 1, ξ i ∈ (0, 1) with 0 < ξ 1 < ξ 2 < · · · < ξ n−2 < 1, and a i , b i satisfy a i , b i ∈ [0, ∞], 0 < n−2 i=1 a i < 1, and n−2 i=1 b i < 1. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple (at least three) positive solutions to the above boundary value problem.  2005 Elsevier Inc. All rights reserved.

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p-Laplacian operator

“…We can see easily that all the results obtained in [1], [2], [4], [6], [7] are the existence of positive solutions. Seeing such a fact, we cannot but ask "how can we find the solutions since the solutions exist definitely?"…”

“…The purpose of this paper is to consider the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the nonlinear two-point singular boundary value problem (BVP) with a p-Laplacian operator (1) (ϕ p (u )) + q(t)f (u) = 0, 0 < t < 1,…”

“…10371006. , and the coefficient q(t) may be singular at t = 0, 1. Several papers have been devoted in the recent years to the study of (1) subject to different linear or nonlinear boundary conditions, see [1], [2], [4], [6], [7] and their references. Here, only positive solutions are meaningful.…”

“…Here, only positive solutions are meaningful. By using the fixed point theorem in cones due to Krasnoselskii [3], Wang [1] and Kong and Wang [2] established the existence of one positive solution for (1) subject to one of the following nonlinear boundary conditions: By applying a new twin fixed point theorem due to Avery and Henderson [5], He and Ge [4] obtained the existence of two positive solutions for (1) subject to (w1), (w2), (w3). By using the fixed point theorem in cones due to Krasnoselskii [3], R. P. Agarwal, Haishen Lü and D. O'Regan [6] studied the problem of eigenvalues of (1) subject to (BCa) when B 0 = B 1 = 0, they also obtained the existence of two positive solutions.…”

Positive solutions for one-dimensional p-Laplacian boundary value problems with nonlinear parameter

“…For details, see for example, Refs [1,2,5], in the case p=2 see [6], and for case λ = 1, see [7,8,9]. In a recent paper [4], Hai considered the boundary value problem ∆u + λa(t)f (u) = 0, t ∈ Ω,…”