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

It is well known that a single nonlinear Schrodinger (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.