Uniqueness and non-uniqueness of least energy normalized solutions for nonlinear Schrödinger equations on compact metric graphs

Abstract

We investigate uniqueness and non-uniqueness of least energy normalized and least energy nodal normalized solutions for nonlinear Schrödinger equations on compact metric graphs. We first prove existence of least energy nodal normalized solutions in the $L^2$-subcritical regime and, at the critical exponent, below a graph-dependent threshold. We then establish a conditional non-uniqueness result for slightly $L^2$-subcritical powers and identify broad classes of graphs for which the required condition either holds or fails. In particular, we prove uniqueness of least energy nodal normalized solutions on the interval for every mass and every $p\in(2,6)$. Finally, using ODE and phase-plane techniques, we show that least energy normalized solutions on the interval are unique for $p$ sufficiently close to $2$. Overall, the results reveal a strong dependence of the uniqueness picture on the nonlinearity power, the topology, and the metric of the graph, and a structural difference between the constant sign and the sign-changing settings.