Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs

We investigate the existence of ground states for the nonlinear Schrodinger Equation on star graphs with two subcritical focusing nonlinear terms: a standard power nonlinearity, and a delta-type nonlinearity located at the vertex. We find that if the standard nonlinearity is stronger than the pointwise one, then ground states exist for small mass only. On the contrary, if the pointwise nonlinearity prevails, then ground states exist for large mass only. All ground states are radial, in the sense that their restriction to each half-line is always the same function, and coincides with a soliton tail. Finally, if the two nonlinearities are of the same size, then the existence of ground states is insensitive to the value of the mass, and holds only on graphs with a small number of half-lines. Furthermore, we establish the orbital stability of the branch of radial stationary states to which the ground states belong, also in the mass regimes in which there is no ground state. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the existence of ground states for the nonlinear Schrodinger equation on star graphs with two subcritical focusing nonlinear terms. The results show that if the standard nonlinearity is stronger than the pointwise nonlinearity, ground states exist only for small mass. Conversely, if the pointwise nonlinearity prevails, ground states exist only for large mass. All ground states are radial and have the same restriction to each half-line, coinciding with a soliton tail. Moreover, when the two nonlinearities are of the same size, the existence of ground states is independent of the mass value and is only valid in graphs with a small number of half-lines.