Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight

In this paper, we shall establish unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that (mu(v)(k), 0) is a bifurcation point of the above problems and there are two distinct unbounded continua, (C-k(v))(+) and (C-k(v))(-), consisting of the bifurcation branch C-k(v) from (mu(v)(k), 0) where mu(v)(k) is the kth positive or negative eigenvalue of the linear problem corresponding to the above problems, v is an element of {+, -}. As the applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. (c) 2013 Elsevier Inc. All rights reserved.