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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

u '''' (x)=λf(u(x)),x(0,1),u(0)=u ' (0)=u(1)=u ' (1)=0,

where λ is a positive parameter and f(u) 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

w '''' (x)=λf ' (u)w,x(0,1),w(0)=w ' (0)=w(1)=w ' (1)=0·

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)].

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE