Existence and Stability of Ground States for the Defocusing Nonlinear Schr\"odinger Equation on Quantum Graphs

We study the existence and stability of ground states for the defocusing nonlinear Schr\"odinger equation on non-compact metric graphs. We establish a sharp criterion for the existence of action ground states, linking it to the spectral properties of the Hamiltonian operator. In particular, we show that ground states exist if and only if the bottom of the spectrum of the Hamiltonian is negative and the frequency lies in a suitable range. Furthermore, we investigate the relationship between action and energy ground states, proving that while every action minimizer is an energy minimizer, the converse may not hold and the energy minimizer may not exist for all masses. To illustrate this phenomenon, we analyze explicit solutions on star graphs with $\delta$ and $\delta'$-couplings, showing that for mass subcritical exponents of the nonlinearity, there exists a large interval of masses for which no energy ground state exists while in the opposite case, they exist for all masses.