Optimizing the first eigenvalue of nonlinear quantum graphs

The main objective of the present work is to study the problem of minimizing or maximizing the first eigenvalue of nonlinear Schrödinger operators on an appropriate specified subset. In the interval case we find that the minimizing and maximizing potentials can extended immediately to the nonlinear Schrödinger operators. In the metric graphs case we show that the maximizing potential on a finite compact metric graphs G is coincide with the maximizing potential on the loop graphs.