Global structure for a fourth-order boundary value problem with sign-changing weight. (English) Zbl 1483.34039
Summary: We study the fourth-order boundary value problem with a sign-changing weight function:
\[
\begin{cases}
u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\quad t\in(0,1),\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{cases}
\]
where \(\lambda\in\mathbb{R}\) is a parameter, \(f_1,f_2\in C([0, 1] \times\mathbb{R}^5,\mathbb{R}),f_1\) is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.
MSC:
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
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