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Global bifurcation and multiple results for Sturm-Liouville problems. (English) Zbl 1223.34030
The following nonlinear Sturm-Liouville boundary value problem
\[ \begin{cases} -(p(t)u'(t))' + q(t)u(t) = \lambda a(t)f(u(t)), \quad 0<t<1, \\ \alpha_1 u(0) + \beta_1 u'(0) = 0, \quad \alpha_2 u(1) + \beta_2 u'(1) = 0\end{cases} \]
is considered, where \(f\): \({\mathbb R} \to {\mathbb R}\) is a continuous function and there exists \(f_0\), \(f_\infty \in (0,\infty)\) such that
\[ f_0 = \lim_{|x|\to0}\, \frac{f(x)}{x}, \quad f_\infty = \lim_{|x|\to\infty} \frac{f(x)}{x}. \]
A global bifurcation result is obtained, and then the existence of solutions having exactly \(k-1\) zeros in \((0,1)\) is derived, where \(k \in {\mathbb N}\).

MSC:
34B24 Sturm-Liouville theory
34C23 Bifurcation theory for ordinary differential equations
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