Sturm-Liouville theory for the radial $\Delta_p$ -operator

Sturm–Liouville Type Problems for the p-Laplacian under Asymptotic Non-resonance Conditions

“…The following Theorem first appeared in Walter [15]. It generalizes a classical and well known theorem for p=2 to general p>1.…”

“…For the standard homogeneous eigenvalue equation (4) uniqueness for the initial value problem was proved by Walter in [15]. Here we present a uniqueness result for the more general homogeneous differential equation (6).…”

“…Local existence follows from Schauder's fixed point theorem as in Walter [15], and global existence follows from Corollary 1. Next we prove uniqueness.…”

Eigenvalues of the p-Laplacian and disconjugacy criteria

“…Let us note that the eigenvalue problem for the p-laplacian has been widely studied in recent years; see, for example, [2,22] among several others, and the references therein. Hence, a characterization for disconjugacy in terms of eigenvalues could be a useful tool.…”

“…In the first case, a solution has an infinite number of isolated zeros; in the second case, a solution has a finite number of zeros. However, from the Sturm-Liouville theory for the p-laplacian ( [11,16,22]; see also the recent monograph [10]) if one solution is oscillatory (resp., nonoscillatory), then every solution is oscillatory (resp., nonoscillatory). Hence, we may speak about oscillatory or nonoscillatory equations, instead of solutions.…”