## 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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### References:

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