# NLS ground states on metric trees: existence results and open questions

@article{Dovetta2019NLSGS, title={NLS ground states on metric trees: existence results and open questions}, author={Simone Dovetta and Enrico Serra and Paolo Tilli}, journal={arXiv: Analysis of PDEs}, year={2019} }

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Polya-Szegő inequality for… Expand

#### 13 Citations

Ground states for the NLS equation with combined nonlinearities on non-compact metric graphs.

- Physics, Mathematics
- 2020

We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$.… Expand

Non-Kirchhoff Vertices and NLS Ground States on graphs

- Physics, Mathematics
- 2020

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrodinger Equation on graphs endowed with vertices where the matching… Expand

Uniqueness and non–uniqueness of prescribed mass NLS ground states on metric graphs

- Mathematics
- 2020

We consider the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schrodinger Energy with power nonlinearity on noncompact metric graphs. We first establish that the… Expand

Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

- Physics
- 2020

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrodinger Equation on graphs endowed with vertices where the matching… Expand

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

- Physics, Mathematics
- 2020

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… Expand

Discontinuous Ground States for NLSE on $\mathbb{R}$ with a F\"{u}l\"{o}p-Tsutsui $\delta$ interaction

- Physics, Mathematics
- 2020

We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schrodinger equation with a focusing power nonlinearity and a defect located at the origin. In this… Expand

Normalized solutions for the Kirchhoff equation on noncompact metric graphs

- Physics
- Nonlinearity
- 2021

In this paper, we are concerned with normalized solutions for the Kirchhoff equation on noncompact metric graphs G . We first study the case that the graph G is the real line and the half line… Expand

Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

- Computer Science, Mathematics
- ArXiv
- 2021

It is shown how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. Expand

Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems

- Mathematics, Computer Science
- ArXiv
- 2019

Using the Łojasiewicz inequality, it is shown that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. Expand

Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs

- Communications on Pure & Applied Analysis
- 2021

<p style='text-indent:20px;'>We consider positive solutions of semi-linear elliptic equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

NLS ground states on graphs

- Mathematics, Physics
- 2014

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states,… Expand

Mass-constrained ground states of the stationary NLSE on periodic metric graphs

- Mathematics
- Nonlinear Differential Equations and Applications NoDEA
- 2019

We investigate the existence of ground states with fixed mass for the nonlinear Schr\"odinger equation with a pure power nonlinearity on periodic metric graphs. Within a variational framework, both… Expand

Negative Energy Ground States for the L2-Critical NLSE on Metric Graphs

- Mathematics, Physics
- 2016

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with L2-critical power nonlinearity on noncompact quantum graphs. We prove that,… Expand

NLS ground states on metric graphs with localized nonlinearities

- Mathematics
- 2016

Abstract We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of… Expand

Bound states of the NLS equation on metric graphs with localized nonlinearities

- Mathematics
- 2016

Abstract We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrodinger equation on a noncompact metric graph. The main feature is that the nonlinearity is… Expand

Ground States of the L 2-Critical NLS Equation with Localized Nonlinearity on a Tadpole Graph

- Mathematics
- 2018

The paper aims at giving a first insight on the existence/nonexistence of ground states for the $L^2$-critical NLS equation on metric graphs with localized nonlinearity. In particular, we focus on… Expand

On the lack of bound states for certain NLS equations on metric graphs

- Mathematics
- 2016

Abstract The purpose of this paper is to prove some results on the absence of bound states for certain nonlinear Schrodinger equations on noncompact metric graphs with localized nonlinearity. In… Expand

$$L^2$$L2-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

- Mathematics, Physics
- Calculus of Variations and Partial Differential Equations
- 2019

Carrying on the discussion initiated in Dovetta and Tentarelli (Ground states of the $$L^2$$L2-critical NLS equation with localized nonlinearity on a tadpole graph, 2018. arXiv:1804.11107 [math.AP]),… Expand

On the spectrum of the Laplacian on regular metric trees

- Mathematics
- 2004

A metric tree Γ is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Δ on such a tree is the operator of second order differentiation on each edge, complemented by the… Expand

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs

- Physics, Mathematics
- Analysis & PDE
- 2019

We investigate the existence of ground states for the focusing nonlinear Schroedinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states… Expand