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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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