Unilateral global bifurcation for fourth-order eigenvalue problems with sign-changing weight

In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν, 0)$, where $μ_k^ν$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in{+,-}$. As the 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.

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