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

This paper concerns the existence of nontrivial solutions for the following singular m-point boundary value problem with a sign-changing nonlinear term [GRAPHICS] where (Lu)(t) = ((p) over tilde (t)u'(t))' + q(t)u(t), 0 < xi(1) < xi(2) < xi(m-2) < 1, a(i) is an element of vertical bar 0. +infinity), h(t) is allowed to be singular at t = 0, 1, and f : vertical bar 0, 1 vertical bar x (-infinity. +infinity) -> (-infinity. +infinity) 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]. (c) 2008 Elsevier B.V. All rights reserved.