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

Abstract

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian

$$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$

subject to the boundary value conditions:

$$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$

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 α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ ⁿ⁻² _i=1 α _i <1 and 0≤∑ ⁿ⁻² _i=1 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.