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

Riccardo Adami, Filippo Boni, Simone Dovetta

Abstract

We investigate the existence of ground states for the nonlinear Schrödinger 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.

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

Abstract

We investigate the existence of ground states for the nonlinear Schrödinger 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.

Paper Structure

This paper contains 7 sections, 13 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. NLSE with standard nonlinearity on the real line
  4. Some remarks on stationary states
  5. Existence criterion and characterization of ground states
  6. Proof of Theorem \ref{['exoutdiag']}--\ref{['exdiag']} and of Proposition \ref{['prop_Np']}
  7. Proof of Proposition \ref{['prop_stab']}

Key Result

Theorem 1.1

Let $p\in(2,6),\,q\in(2,4)$ and $q\neq\frac{p}{2}+1$. Then there exists a critical mass $\mu_{p,q}>0$ such that Furthermore, whenever they exist, ground states at prescribed mass are unique and they are radial and decreasing on $S_N$, in the sense that their restriction to each half--line of the graph corresponds to the same decreasing function on $\mathbb{R}^+$.

Figures (2)

  • Figure 1: The stationary states of Proposition \ref{['prop_stat']} on $S_5$: $\eta_0^\omega$ (A), $\eta_1^\omega$ (B) and $\eta_2^\omega$ (C).
  • Figure 2: The graph of $R(p):=\frac{\mathcal{I}\left(\sqrt{\frac{p}{p+18}}\right)}{\mathcal{I}(0)}$ as a function of $p\in(2,6)$. The validity of \ref{['condcrit']} at $N=3$ is equivalent to $R(p)\leq0$. These numerical simulations suggest that condition \ref{['condcrit']} always holds at $N=3$, for every $p\in(2,6)$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 16 more