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Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. (English) Zbl 1060.34014

This paper deals with the existence and exact multiplicity of positive solutions of the following fourth-order Dirichlet boundary value problem

$\begin{array}{cc}& {u}^{\text{'}\text{'}\text{'}\text{'}}\left(x\right)=\lambda f\left(u\left(x\right)\right),\phantom{\rule{1.em}{0ex}}x\in \left(0,1\right),\hfill \\ & u\left(0\right)={u}^{\text{'}}\left(0\right)=u\left(1\right)={u}^{\text{'}}\left(1\right)=0,\hfill \end{array}$

where $\lambda$ is a positive parameter and $f\left(u\right)$ 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

$\begin{array}{cc}& {w}^{\text{'}\text{'}\text{'}\text{'}}\left(x\right)=\lambda {f}^{\text{'}}\left(u\right)w,\phantom{\rule{1.em}{0ex}}x\in \left(0,1\right),\hfill \\ & w\left(0\right)={w}^{\text{'}}\left(0\right)=w\left(1\right)={w}^{\text{'}}\left(1\right)=0·\hfill \end{array}$

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)].

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE