近年来，具有物理学背景的偏微分方程及方程组在数学界引起广泛关注，众 多数学和物理工作者对非线性椭圆方程及方程组的解的存在性、唯一性、多解性 等性质展开了深入研究，并得到了大量优秀的成果。由于结构更加复杂，目前关于?-耦合非线性椭圆方程组的结论相对较少，方程组非平凡解的存在性、唯一性、多解性、渐近收敛行为等方面仍有许多重要的问题有待解决。系数方程组的解的存在性也为方程组的解的分类刻画带来了困难。本文旨在通过经典的变分法和椭圆方程理论，研究非线性椭圆方程组的极小能量估计和非平凡解的存在性等相关性质。首先我们考虑空间维数? ⩾ 5 的临界?-耦合非线性薛定谔方程组。我们利用数学归纳法，对非线性薛定谔方程组的极小能量进行了更加精细的估计，从而得到当耦合系数全部大于零时，临界?-耦合方程组及对应的全空间上的极限方程组存在正的基态解。我们通过巧妙的构造，将方程组系数满足的代数方程组的正解存在性问题转化为极小值存在性问题，首次简洁而完整地证明了代数方程组正解的存在性，并进一步刻画出基态解的形态。该结果推广了Chen, Luo, Wu 和Zou 等关于方程组基态解存在性的结果，并首次给出了空间维数? ⩾ 5 的临界?-耦合非线性薛定谔方程组基态解的分类结果。另外，我们将上述结果推广到空间维数? > 4𝑠 的带分数阶Laplace 的临界?-耦合非线性椭圆方程组。我们系统性地研究了方程组基态解的存在性和非存在性等性质。由于含分数阶Laplace 的非线性方程组更加复杂，我们在证明过程中将利用一些新技巧。此外，我们关注方程组的解的渐近收敛行为，并将Chen 和Zou的相关结果推广到带分数阶Laplace 的非线性椭圆方程组，同时得到另一类方程组的正的基态解的存在性。最后，我们考虑非线性耦合薛定谔方程组的正规化解问题。关于方程组的正规化解结果相对较少。正规化解的存在性可以转化为寻找能量泛函在限制下的临界点。我们研究的方程组可以看作是玻色爱因斯坦凝聚型方程组的线性扰动，但是线性扰动项的存在使得验证极小化序列的紧性变得更加困难。我们通过Shibata 重排和Schwarz 对称单调递减重排得到极小化序列的收敛性，进一步证明了方程组正规化解的存在性。

In recent years, partial differential equations and systems which originated from physical problems have received much attention from science researchers. Many mathematicians and physicians have deeply investigated in nonlinear elliptic equations and systems and plenty of excellent results have come out successively, for example, existence, uniqueness and multiplicity of solutions to the systems.Heretofore there are few results concerning Bose-Einstein condensate related ?-coupled systems due to the complicated structure. Existence, uniqueness, multiplicity, asymptotic behaviour and other qualitative results of nontrivial solutions to the ?-coupled systems remain to be investigated. Meanwhile, much difficulties arises from the existence of parameters related algebraic systems when classifying the solutions. In this thesis, we aim to study the existence and qualitative results of nontrivial solutions, as well as the the least energy estimates of the Bose-Einstein condensate related systems, via variational methods and classical elliptic equations theories.Firstly, we consider the critical ?-coupled nonlinear Schr?dinger systems in dimension ? ⩾ 5. We introduce the idea of induction to get the refined estimates of the least energy and obtain the existence results of positive ground state solutions to the critical ?-coupled nonlinear Schr?dinger systems and related limit systems if all coupling constants are positive. Meanwhile, we give the first answer to the existence of positive solutions to the algebraic system and subsequently illustrate the ground state solutions of synchronized type, where an ingenious idea is used to transform the existence of positive solutions of the algebraic systems to the existence of infimum of a geometric problem. These results generalize those of Chen, Luo, Wu and Zou, and give the first classification results of positive ground state solutions of the the critical ?-coupled nonlinear Schr?dinger systems in dimension ? ⩾ 5.Besides, we generalize these results to the the critical ?coupled nonlinear Schr?dinger systems involving fractional Laplace in dimension ? > 4𝑠. We systematically investigate in the existence and nonexistence of ground state solutions. Since the nonlinear systems involving fractional Laplace are more complicated, we shall introduce some new techniques. Meanwhile, we are interested in the asymptotic behaviour of the solutions. We generalize the results of Chen and Zou to the systems involving fractional Laplace and obtain the existence results of another system at the same time.Finally, we consider the normalized solutions to the nonlinear coupled Schr?dinger systems. There are few results involving systems. The existence of normalized solutions can be obtained by studying the critical points of energy functional under some constraints. The systems can be treated as a linear perturbation of BEC systems, which arises much difficulties when dealing with the compactness of minimizing sequences. We obtain the compactness of minimizing sequences via Shibata’s rearrangement and Schwarz’s rearrangement, and further we prove the existence of normalized solutions.