Existence and multiplicity of positive solutions for two couplednonlinear Schrödinger equations

Tai-Chia Lin; Tsung-Fang Wu

doi:10.3934/dcds.2013.33.2911

Article Contents

2013, Volume 33, Issue 7: 2911-2938. Doi: 10.3934/dcds.2013.33.2911

This issue Previous Article Partial hyperbolicity and central shadowing Next Article The splitting lemmas for nonsmooth functionals on Hilbert spaces I

Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations

Tai-Chia Lin^1, and
Tsung-Fang Wu^2,

1.
Department of Mathematics & National Center for Theoretical Sciences at Taipei, National Taiwan University, Taipei, 10617
2.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811

Received: March 31, 2012

Revised: September 30, 2012

Published: June 2013

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Abstract

It is well known that a single nonlinear Schrödinger (NLS) equation with a potential $V$ and a small parameter $\varepsilon $ may have a unique positive solution that is concentrated at the nondegenerate minimum point of $V$ . However, the uniqueness may fail for two-component systems of NLS equations with a small parameter $\varepsilon $ and potentials $V_{1}$ and $V_{2}$ having the same nondegenerate minimum point. In this paper, we will use energy estimates and category theory to prove the nonuniqueness theorem.

Keywords:

Two coupled nonlinear Schrödinger equations,
Lusternik-Schnirelmann category,
multiple positive solutions.

Mathematics Subject Classification: Primary: 35J50, 35J57; Secondary: 35J47.

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