Abstract
By a node of a Sturm–Liouville problem, it means an interior zero of an eigenfunction. In this paper, by considering the unique node of the second Dirichlet eigenfunction as a nonlinear functional of potential from the Lebesgue space , we will study the optimization problems to minimize or to maximize subject to the constraint . By applying the recent results on the differentiability and complete continuity of in , it will be proved that for the case , these optimization problems are attained by some potentials. Moreover, a critical equation for optimizers will be derived. Finally, by considering the limit case , it will be found that the optimizers for the optimization problems for the case are certain Dirac measures. These results are then used to deduce the optimal locations of nodes .
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