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Zbl 1060.34014
Korman, Philip
Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. (English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 1, 179-190 (2004). ISSN 0308-2105; ISSN 1473-7124/e

This paper deals with the existence and exact multiplicity of positive solutions of the following fourth-order Dirichlet boundary value problem $$ \align & u''''(x) = \lambda f(u(x)),\quad x\in(0,1),\\ & u(0) = u'(0) = u(1) = u'(1) = 0, \endalign $$ where $\lambda$ is a positive parameter and $f(u)$ is a continuous function. The main tool is the bifurcation theory, in particular, the bifurcation results of {\it M. G. Crandall} and {\it 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 $$ \align & w''''(x) = \lambda f'(u)w,\quad x\in(0,1),\\ & w(0) = w'(0) = w(1) = w'(1) = 0. \endalign $$ The author investigates this linearized problem by using the classical paper of {\it W. Leighton} and {\it 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)].
[Petr Necesal (Plzen)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE

Keywords: bifurcation of solutions; positive solutions; fourth-order Dirichlet problem

Citations: Zbl 0275.47044; Zbl 0084.08104; Zbl 1011.35046

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