This paper deals with the existence and exact multiplicity of positive solutions of the following fourth-order Dirichlet boundary value problem
where is a positive parameter and is a continuous function. The main tool is the bifurcation theory, in particular, the bifurcation results of M. G. Crandall and P. H. Rabinowitz [Arch. Ration. Mech. Anal. 52, 161-180 (1973; Zbl 0275.47044)] are used. The most difficult part of the used approach is to show the positivity of the nontrivial solutions of the corresponding linearized problem
The author investigates this linearized problem by using the classical paper of W. Leighton and Z. Nehari [Trans. Am. Math. Soc. 89, 325–377 (1959; Zbl 0084.08104)]. The previous known results for the nonlinear problem were obtained by using shooting techniques, Leray-Schauder degree theory together with monotone iterations. The bifurcation approach was also applied to a similar problem but in the case of different boundary conditions in another paper of the author [Math. Methods Appl. Sci. 25, No. 1, 3–20 (2002; Zbl 1011.35046)].