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

Abstract

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 μkν,0 is a bifurcation point of the above problems and there are two distinct unbounded continua, Ckν+ and Ckν-, consisting of the bifurcation branch Ckν from μkν,0, where μkν is the kth positive or negative eigenvalue of the linear problem corresponding to the above problems, ν∈{+,-}. 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.