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Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight. (English) Zbl 1294.34017

Summary: In this paper, we shall establish unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on the perturbation function, we show that \((\mu_k^{\nu},0)\) is a bifurcation point of the above problems and there are two distinct unbounded continua, \((\mathcal C_k^{\nu})^+\) and \((\mathcal C_k^{\nu})^-\), consisting of the bifurcation branch \(\mathcal C_k^{\nu}\) from \(({\mu}_k^{\nu},0)\), where \(\mu_k^{\nu}\) is the \(k\)th positive or negative eigenvalue of the linear problem corresponding to the above problems, \(\nu\in \{+,-\}\). As applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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