Multiple solutions of impulsive Sturm–Liouville boundary value problem via lower and upper solutions and variational methods

Abstract

In this paper, we prove the existence of multiple solutions for second order Sturm–Liouville boundary value problem { − L u = f ( x , u ) , x ∈ [ 0 , 1 ] \\ { x 1 , x 2 , … , x l } , − Δ ( p ( x i ) u ′ ( x i ) ) = I i ( u ( x i ) ) , i = 1 , 2 , … , l , R 1 ( u ) = 0 , R 2 ( u ) = 0 , where L u = ( p ( x ) u ′ ) ′ − q ( x ) u is a Sturm–Liouville operator, R 1 ( u ) = α u ′ ( 0 ) − β u ( 0 ) , R 2 ( u ) = γ u ′ ( 1 ) + σ u ( 1 ) . The technical approach is fully based on lower and upper solutions and variational methods. The interesting point is that the property that the critical points of the energy functional are exactly the fixed points of an operator that involves the Greenʼs function. Besides, the existence of four solutions is given.