# American Institute of Mathematical Sciences

July  2013, 33(7): 2911-2938. doi: 10.3934/dcds.2013.33.2911

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

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

Received  April 2012 Revised  October 2012 Published  January 2013

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.
Citation: Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
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