Nonlinear Schrödinger Equations on Periodic Metric Graphs

Alexander Pankov

doi:10.3934/dcds.2018030

Article Contents

2018, Volume 38, Issue 2: 697-714. Doi: 10.3934/dcds.2018030

This issue Previous Article Nonexistence results for elliptic differential inequalities with a potential in bounded domains Next Article The continuum limit of Follow-the-Leader models — a short proof

Nonlinear Schrödinger Equations on Periodic Metric Graphs

Alexander Pankov^1,2,

1.
Mathematics Department, Morgan State University, Baltimore, MD 21251, USA
2.
RUDN University, Moscow 117198, Russia

Received: June 2017

Revised: August 2017

Published: November 2017

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Abstract

The paper is devoted to the nonlinear Schrödinger equation with periodic linear and nonlinear potentials on periodic metric graphs. Assuming that the spectrum of linear part does not contain zero, we prove the existence of finite energy ground state solution which decays exponentially fast at infinity. The proof is variational and makes use of the generalized Nehari manifold for the energy functional combined with periodic approximations. Actually, a finite energy ground state solution is obtained from periodic solutions in the infinite wave length limit.

Keywords:

Metric graph,
periodic nonlinear Schrödinger equation,
generalized Nehari manifold,
periodic approximation.

Mathematics Subject Classification: Primary: 35Q55, 35R02; Secondary: 49J40, 58E30.

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