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Exact multiplicity results for boundary value problems with nonlinearities generalising cubic. (English) Zbl 0855.34022
The authors establish two exact multiplicity results for the boundary value problem of the form \(u'' + \lambda f(x,u) = 0\), \(x \in [-1,1]\), \(u(-1) = u(1) = 0\), \(\lambda > 0\), where \(f\) behaves like a cubic in \(u\). The model problems are the following: \(\text{(BVP}_0)\): \(f(x,u) = u^2 (b(x) - u)\) with positive even \(b(x)\) such that \(b'(x)\), \(b''(x) < 0\), \(b''(x) \leq 0\) \(\forall x \in (0,1]\), and \(\text{(BVP}_1)\): \(f(u) = (u - a) (u - b) (c - u)\) with \(0 < a < b < c\) and \(\int^c_a f(u) du > 0\). Under some additional technical assumptions, the authors show the existence of \(\lambda_i\) such that \(\text{BVP}_i\), \(i = 0,1\) has exactly \(i\) positive solutions for \(0 < \lambda < \lambda_i\), exactly \(i + 1\) positive solutions for \(\lambda = \lambda_i\), and exactly \(i + 2\) positive solutions for \(\lambda > \lambda_i\). This paper is of particular interest because of the use of techniques of bifurcation theory. In particular, a bifurcation theorem of M. G. Crandall and P. H. Rabinowitz [Arch. Rat. Mech. Analysis 52, 161-180 (1973; Zbl 0275.47044)] is involved essentially in the analysis of \(\text{BVP}_1\).

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:
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