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Nontrivial solutions of singular fourth-order Sturm-Liouville boundary value problems with a sign-changing nonlinear term. (English) Zbl 1213.34045

Summary: This paper concerns the existence of nontrivial solutions for the following singular boundary value problem with a sign-changing nonlinear term:
\[ \begin{cases} u^{(4)}(t) =h(t)f(t,u(t),u''(t)),\quad 0<t<1,\\ \alpha_1u(0)-\beta_1u'(0)=\delta_1u(1)+\gamma_1u'(1)=0,\\ \alpha_2u''(0)-\beta_2u'''(0)=\delta_2u''(1)+\gamma_2u'''(1)=0,\end{cases} \]
where \(h(t)\) is allowed to be singular at \(t = 0\) and/or \(t=1\). Moreover, \(f(t,x,y):[0,1]\times \mathbb R^2\to\mathbb R\) is a sign-changing continuous function and may be unbounded from below with respect to \(x\) and \(y\). By applying the topological degree of a completely continuous field and eigenvalue, some new results on the existence of nontrivial solutions for the above boundary value problem are obtained.

MSC:

34B24 Sturm-Liouville theory
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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