Optimization problems on nodes of Sturm–Liouville operators with \(L^p\) potentials

The aim of this paper is to obtain the optimal characterizations of locations for all nodes of the classical Sturm-Liouville operators, given the \(L^p\) $L^p$ norms with \(1<p <\infty \) $1<p<\infty$ of the potentials. Regarding the ith node of the mth eigenfunction as a functional of the potential, we deduce critical equations to determine the minimizing potential such that the node is minimized. From the critical equations, we obtain two equivalent characterizations of the minimal nodes, which are written as nonlinear systems for 4-dimensional or 2-dimensional parameters. These optimal characterizations can yield the sharp lower and upper bounds for the locations of nodes.