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Global behavior of positive solutions for some semipositone fourth-order problems. (English) Zbl 1448.34062

Summary: In this paper, we study the global behavior of positive solutions of fourth-order boundary value problems \[ \begin{cases} u''''=\lambda f(x,u),\quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0, \end{cases} \] where \(f: [0,1]\times \mathbb{R^{+}} \rightarrow \mathbb{R}\) is a continuous function with \(f(x,0)<0\) in \((0, 1)\), and \(\lambda >0\). The proof of our main results are based upon bifurcation techniques.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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