Multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one-dimensional [formula omitted]-Laplacian

Abstract

In this paper we consider the multipoint boundary value problem for the one-dimensional p -Laplacian ( ϕ p ( u ′ ( t ) ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to the boundary conditions u ( 0 ) = ∑ i = 1 n μ i u ( ξ i ) , u ( 1 ) = ∑ i = 1 n μ i u ( η i ) , where ϕ p ( s ) = | s | p − 2 s , p > 1 , μ i ≥ 0 , 0 ≤ ∑ i = 1 n μ i < 1 , 0 < ξ 1 < ξ 2 < ⋯ < ξ n < 1 / 2 , ξ i + η i = 1 , i = 1 , 2 , … , n . Applying a fixed point theorem of functional type in a cone, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is that the nonlinear term f contains the first-order derivative explicitly.