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

TL;DR

This paper studies how competing nonlinearities in the nonlinear Schrödinger equation on star graphs influence the existence and stability of ground states, revealing threshold phenomena depending on the dominance of nonlinear terms.

Contribution

It introduces a detailed analysis of ground state existence and stability in NLS equations with competing nonlinearities on star graphs, highlighting threshold effects based on nonlinearity strength.

Findings

Ground states exist only for small mass if standard nonlinearity dominates.

Ground states exist only for large mass if pointwise nonlinearity dominates.

Radial ground states are stable and resemble soliton tails.

Abstract

We investigate the existence of ground states for the nonlinear Schr\"odinger 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.