Weighted \(p\)-Laplacian problems on a half-line (Q890195)

Generalizing a Sturm-Liouville setting (\(p = 2\)), the authors consider the \(p\)-Laplacian problem \[ -(|y'|^{p-1} \mathrm{sgn} y')' = (p-1) (\lambda r -q) |y|^{p-1} \mathrm{sgn} y \text{ on } [0,\infty) \text{ for } 1<p<\infty \] subject to the initial condition \(y'(0) \sin \alpha = y(0) \cos \alpha\) \((\alpha \in [0, \pi))\). In a first step, it is assumed that \(r(x) > 0\) a.e. Then, \(\lambda \in {\mathbb R}\) is called an eigenvalue if there is a non-trivial solution (``eigenfunction'') \(y\) with \(\int_0^\infty r |y|^p < \infty\). Under certain conditions on \(q\) and \(r\) (or, more precisely, on \(s(x) := r(x)^{1/p}\)) a modified Prüfer angle \(\phi(\lambda, x)\) is introduced. In this case, as the first main result, the existence of a strictly increasing sequence of eigenvalues \((\lambda_n)\) is shown such that \(\lambda_n \rightarrow \infty\) for \(n \rightarrow \infty\) and \(\phi(\lambda_n, x) \rightarrow (n+1) \pi_p -\) for \(x \rightarrow \infty\) and \(\phi(\lambda, x) \rightarrow (n+1) \pi_p +\) for \(x \rightarrow \infty\), \(\lambda \in (\lambda_n, \lambda_{n+1})\) where \(\pi_p = 2 \pi/(p \sin \frac{\pi}{p})\). Here, \(-\) and \(+\) shall indicate that the limit is from below or above, respectively. It is observed that each eigenvalue \(\lambda_n\) has an eigenfunction \(y_n\) with precisely \(n\) zeros in \((0,\infty)\). In the second step, this result is generalized to a setting where the weight function \(r\) may have sign changes (i.e., \(r(x) > 0\) and \(r(x) < 0\) both on sets of non-zero measure) and where \(r\) may vanish on sets of non-zero measure. This situation is more delicate since the eigenvalues may accumulate at \(+ \infty\) as well as at \(- \infty\) and the setting \(s = r^{1/p}\) must be replaced by an inequality of the form \(|r| \leq s^{p}\). The key tool is the consideration of the two parameter equation \(-(|y'|^{p-1} \mathrm{sgn} y')' = (p-1) (\lambda r - \mu s^p - q) |y|^{p-1} \mathrm{sgn} y\). Most of the results of the second step seem to be new also for the Sturm-Liouville case \(p = 2\).