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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 $$\cases -(p(t)u'(t))' + q(t)u(t) = \lambda a(t)f(u(t)), \quad 0<t<1, \\ \alpha_1 u(0) + \beta_1 u'(0) = 0, \quad \alpha_2 u(1) + \beta_2 u'(1) = 0\endcases$$ is considered, where $f$: ${\Bbb R} \to {\Bbb R}$ 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 $k \in {\Bbb N}$.

34B24Sturm-Liouville theory
34C23Bifurcation (ODE)
Full Text: DOI
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