Non-definite Sturm-Liouville problems for the \(p\)-Laplacian (Q2890633)

The classical indefinite Sturm-Liouville problem NEWLINE\[NEWLINE-y'' + q y = \lambda r y \quad\mathrm{on } [0,1]NEWLINE\]NEWLINE with boundary conditions NEWLINE\[NEWLINEy'(j) \sin \alpha_j = y(j) \cos \alpha_j \quad (j=0,1)NEWLINE\]NEWLINE and with real functions \(r, q \in L_1(0,1)\) can be generalized in various ways.NEWLINENEWLINEIn the present paper, the differential equation is replaced by NEWLINE\[NEWLINE-\Delta_p y = (p - 1) (\lambda r - q) [y]^{p - 1},NEWLINE\]NEWLINE where \(p > 1\), \([y]^{p - 1} := |y|^{p - 1} \sin y\) and \(\Delta_p\) is the \(p\)-Laplacian \(\Delta_p y := ([y']^{p - 1})'\) such that the classical situation corresponds to \(p = 2\). The asymptotics of the eigenvalues and their dependence on the parameters are studied. Furthermore, the oscillation of the eigenfunctions and the interlacing of zeroes is investigated. Different results are obtained for the completely indefinite case and for the right and the left (semidefinite) cases (R(S)D/L(S)D). Here, R(S)D and L(S)D are generalizations from the classical setting. Indeed, for \(p = 2\) the meaning corresponds to the (semi-)definiteness of the quadratic forms associated with the left or the right hand side of the differential equation in the classical form. As a particular difference in the RSD case, there appears one sequence of eigenvalues (with increasing oscillation count) accumulating at \(+ \infty\) or at \(- \infty\), whereas, if RSD fails, there are at least two such sequences, one accumulating at \(+ \infty\) and one at \(- \infty\). If additionally we have LD (or some modified version of LD), these two sequences are unique and the results can be sharpened. The key tool thoughout the paper is an extended version of the Prüfer angle. Most of the results are new for \(p \neq 2\) and some even for the case \(p = 2\).