Nontrivial solutions of m-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term

Abstract

This paper concerns the existence of nontrivial solutions for the following singular m -point boundary value problem with a sign-changing nonlinear term { ( L u ) ( t ) + h ( t ) f ( t , u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , where ( L u ) ( t ) = ( p ̃ ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 , a i ∈ [ 0 , + ∞ ) , h ( t ) is allowed to be singular at t = 0 , 1 , and f : [ 0 , 1 ] × ( − ∞ , + ∞ ) → ( − ∞ , + ∞ ) is a sign-changing continuous function and may be unbounded from below. By applying the topological degree of a completely continuous field and the first eigenvalue and its corresponding eigenfunction of a special linear operator, some new results on the existence of nontrivial solutions for the above singular m -point boundary value problem are obtained. An example is then given to demonstrate the application of the main results. The work improves and generalizes the main results of [G. Han, Y. Wu, Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms, J. Math. Anal. Appl. 325 (2007) 1327–1338; J. Sun, G. Zhang, Nontrivial solutions of singular superlinear Sturm-Liouville problem, J. Math. Anal. Appl. 313 (2006) 518–536].