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

TL;DR

This paper establishes unilateral global bifurcation results for fourth-order eigenvalue problems with sign-changing weights, identifying bifurcation points and unbounded solution continua, and applies these to nodal solutions and comparison theorems.

Contribution

It introduces new bifurcation results for fourth-order problems with sign-changing weights, including the existence of unbounded solution branches and a Sturm type comparison theorem.

Findings

Identified bifurcation points at eigenvalues of the linear problem.

Proved existence of two distinct unbounded continua of solutions.

Established a Sturm type comparison theorem for fourth-order problems.

Abstract

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 $(\mu_k^\nu,0)$ 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 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.