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: ϕ p ( u ′ ( 0 ) ) = ∑ i = 1 n − 2 a i ϕ p ( u ′ ( ξ i ) ) , u ( 1 ) = ∑ i = 1 n − 2 b i u ( ξ i ) , 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 < ∑ i = 1 n − 2 a i < 1 , and ∑ i = 1 n − 2 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.