Sign-changing and multiple solutions of the Sturm–Liouville boundary value problem via invariant sets of descending flow

Abstract

In this paper, we prove the existence of sign-changing and multiple solutions for the second-order Sturm–Liouville boundary value problem { − L u = f ( x , u ) , x ∈ [ 0 , 1 ] R 1 ( u ) = 0 , R 2 ( u ) = 0 , where L u = ( p ( x ) u ′ ) ′ − q ( x ) u is the Sturm–Liouville operator, R 1 ( u ) = α u ′ ( 0 ) − β u ( 0 ) and R 2 ( u ) = γ u ′ ( 1 ) + σ u ( 1 ) . The technical approach is fully based on minimax methods and invariant sets of descending flow.