Synchronization of positive solutions for coupled Schrödinger equations

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In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation $\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$ where $ 2< p<\frac{n}{n-2}, $ if $ n\ge 3$ and $ 2< p<+∞ $, if $ n = 1, 2, $ and $μ_1, μ_2, β>0 $ are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case $ n = 1 $ by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case $ p = 2 $ for which uniqueness of positive solutions was known ([19]).