In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

中文翻译：

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非线性 Schrödinger-Korteweg-de Vries 型系统的多个非平凡解的存在性

在本文中，我们关注非线性薛定谔-Korteweg-de Vries型系统（NLS-NLS-KdV）非平凡解的存在性、不存在性和分岔。首先，我们找到了一些条件来保证系统正解的存在和不存在。其次，我们研究了正基态解的渐近行为。最后，我们使用经典的 Crandall-Rabinowitz 局部分岔理论得到非平凡正解。为了得到这些结果，我们遇到了一些新的挑战。通过结合 Nehari 流形约束方法和精细的能量估计，我们克服了困难，从一个半平凡解中找到了两个分岔分支。这是一个新的有趣现象，但以前没有发现。