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On the global set of solutions of a nonlinear ODE: theoretical and numerical description. (English) Zbl 0597.34014
Nonlinear differential equations of the type $$-u''=f(u)+\lambda g(x)$$ in (0,1), $$u(0)=u(1)=0$$ are studied. Here f is a convex, nondecreasing function with $$f(0)=0$$, $$f'(0)=0$$, $$\lim_{s\to \pm \infty}f(s)/s=\infty$$ and g is a positive function in (0,1). In the particular case, $$f(u)=u| u|^{p-1}$$ and g(x)$$\equiv 1$$, a complete global description of the set of solutions is given exhibiting an infinite number of bifurcation points and turning points. Some of the results regarding the positive solutions are shown to extend for a similar semilinear elliptic problem in higher dimensions.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems, general theory 34C99 Qualitative theory for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations
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