Classification of second-order linear differential equations and an application to singular elliptic eigenvalue problems (Q2893271)

The author studies the Sturm-Liouville equation NEWLINE\[NEWLINEu''+\lambda p(t) u=0,\tag{1}NEWLINE\]NEWLINE where \(\lambda>0\) is a parameter, with attention to the case when (1) is conditionally oscillatory, i.e., there exists a \(\bar\lambda>0\) such that (1) is nonoscillatory for \(0<\lambda<\bar \lambda\) and oscillatory for \(\lambda>\bar \lambda\). Results are obtained for the asymptotic behavior and the number of zeros of the solutions. In particular, criteria are established for the oscillation/nonoscillation of the equation when \(\lambda=\bar \lambda\). These results are also applied to singular elliptic eigenvalue problems on a ball in \({\mathbb{R}}^n\). This work provides an interesting development of the classical oscillation theory for second-order linear differential equations.