Skip to main content Accessibility help
×
Home

Local minimizers in absence of ground states for the critical NLS energy on metric graphs

Abstract

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

Copyright

References

Hide All
1Adami, R., Serra, E. and Tilli, P.. NLS ground states on graphs. Calc. Var. Partial Differ. Equ. 54 (2015), 743761.
2Adami, R., Serra, E. and Tilli, P.. Threshold phenomena and existence results for NLS ground states on metric graphs. J. Funct. Anal. 271 (2016), 201223.
3Adami, R., Serra, E. and Tilli, P.. Negative energy ground states for the L 2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387406.
4Adami, R., Serra, E. and Tilli, P.. Nonlinear dynamics on branched structures and networks. Riv. Math. Univ. Parma (N.S.) 8 (2017), 109159.
5Adami, R., Serra, E. and Tilli, P.. Multiple positive bound states for the subcritical NLS equation on metric graphs. Calc. Var. Partial Differ. Equ. 58 (2019), 5.
6Bellazzini, J., Boussaïd, N., Jeanjean, L. and Visciglia, N.. Existence and stability of standing waves for supercritical NLS with a partial confinement. Comm. Math. Phys. 353 (2017), 229251.
7Bellazzini, J., Georgiev, V. and Visciglia, N.. Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension. Math. Ann. 371 (2018), 707740.
8Bellazzini, J. and Jeanjean, L.. On dipolar quantum gases in the unstable regime. SIAM J. Math. Anal. 48 (2016), 20282058.
9Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), 313345.
10Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), 347375.
11Berkolaiko, G. and Kuchment, P. 2013 Introduction to quantum graphs, Vol. 186, Mathematical Surveys and Monographs. RI, Providence: American Mathematical Society.
12Cazenave, T. 2003 Semilinear Schrödinger equations, Vol.10, Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences.
13Dovetta, S.. Existence of infinitely many stationary solutions of the L 2-subcritical and critical NLSE on compact metric graphs. J. Differ. Equ. 264 (2018), 48064821.
14Dovetta, S.. Mass-constrained ground states of the stationary NLSE on periodic metric graphs. NoDEA Nonlinear Differ. Equ. Appl. 26 (2019), 30.
15Dovetta, S. and Tentarelli, L.. L 2-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features. Calc. Var. Partial Differ. Equ. 58 (2019), 108.
16Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74 (1987), 160197.
17Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94 (1990), 308348.
18Noja, D.. Nonlinear Schrödinger equation on graphs: recent results and open problems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), 20130002.
19Noja, D. and Pelinovsky, D. E.. Standing waves of the quintic NLS equation on the tadpole graph. Preprint arXiv:2001.00881, 2020.
20Noris, B., Tavares, H. and Verzini, G.. Existence and orbital stability of the ground states with prescribed mass for the L 2-critical and supercritical NLS on bounded domains. Anal. PDE 7 (2014), 18071838.
21Noris, B., Tavares, H. and Verzini, G.. Stable solitary waves with prescribed L 2-mass for the cubic Schrödinger system with trapping potentials. Discrete Contin. Dyn. Syst. 35 (2015), 60856112.
22Noris, B., Tavares, H. and Verzini, G.. Normalized solutions for nonlinear Schrödinger systems on bounded domains. Nonlinearity 32 (2019), 10441072.
23Pierotti, D. and Verzini, G.. Normalized bound states for the nonlinear Schrödinger equation in bounded domains. Calc. Var. Partial Differ. Equ. 56 (2017), 133.
24Serra, E. and Tentarelli, L.. Bound states of the NLS equation on metric graphs with localized nonlinearities. J. Differ. Equ. 260 (2016), 56275644.
25Soave, N.. Normalized ground states for the NLS equation with combined nonlinearities. Preprint arXiv:1811.00826, 2018.
26Soave, N.. Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. Preprint arXiv:1901.02003, 2019.

Keywords

MSC classification

Local minimizers in absence of ground states for the critical NLS energy on metric graphs

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.