The paper is concerned with the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_j+ \lambda u_j=\mu |u_j|^{2p}u_j+\sum _{i\ne j}^m \beta |u_i|^{p+1}|u_j|^{p-1}u_j, &{} \text {in}\ \mathbb {R}^n, \\ u_j(x)\rightarrow 0\ \text {as}\ |x|\ \rightarrow \infty , \quad j=1,2,\ldots , m, \end{array}\right. \end{aligned}$$

where \( m\ge 2\), \(0<p<\frac{2}{(n-2)^+}\), \(\lambda >0,\) \(\mu >0\) and \(\beta >0\). We establish a sufficient and necessary condition for the existence of nontrivial ground-state solutions which have the least energy among all the non-zero solutions of the system and whose components have the same modulus. This gives an affirmative answer to a conjecture raised in Correia (Nonlinear Anal. 140:112–129, 2016).

中文翻译：

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耦合薛定ding系统的基态分类

本文关注的是耦合非线性Schrödinger系统非平凡基态解的存在

$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta u_j + \ lambda u_j = \ mu | u_j | ^ {2p} u_j + \ sum _ {i \ ne j} ^ m \ beta | u_i | ^ {p + 1} | u_j | ^ {p-1} u_j，＆{} \ text {in} \ \ mathbb {R} ^ n，\\ u_j（x）\ rightarrow 0 \ \ text { as} \ | x | \ \ rightarrow \ infty，\ quad j = 1,2，\ ldots，m，\ end {array} \ right。\ end {aligned} $$

其中\（m \ ge 2 \），\（0 <p <\ frac {2} {（n-2）^ +} \），\（\ lambda> 0，\） \（\ mu> 0 \）和\（\ beta> 0 \）。我们为系统中所有非零解中能量最小的非平凡基态解的存在建立了充分必要的条件。这为在Correia中提出的猜想提供了肯定的答案（Nonlinear Anal。140：112–129，2016）。