椭圆型方程在凝聚态物理、非线性光学、流体力学、弹性力学、经济学和非线性扩散等领域有着十分广泛的应用。关于椭圆型方程解的相关问题研究一直是非线性偏微分方程领域的热点之一，众多数学家们对其做出了一系列杰出的工作。本文主要通过变分法与椭圆方程理论分别对几类重要的椭圆型方程（组）解的存在性和相关性质进行研究。首先，我们研究了含 Sobolev 临界指数的非线性椭圆方程组的正规化解，其中该方程组所对应的标量方程的非线性项有两项。在质量超临界情形下，我们用新的技巧证明了标量方程的基态能量关于质量的单调非增性，并结合集中紧性原理和精细估计，克服了由临界指数导致的紧性缺失的困难，证明了正规化解的存在性。这是该方程组在 3 维或 4 维情形下的第一个结果。然后，对于混合情形，我们通过对方程组的 Pohozaev 流形进行分解，证明了能量泛函存在局部极小值点并且该局部极小值点还是基态解。此外，我们采用反证法，结合 Hadamard 三球定理和特征值理论，得到了部分非存在性结果。其次，我们研究了一类带位势的分数阶椭圆方程的正规化解。利用变分法，我们将正规化解存在性问题转化成方程相应泛函的极小化序列收敛性问题。基于迭代技巧，我们建立了关于L2-约束极小化问题的次可加不等式新的证明，克服了由位势项和具有非局部特征的分数阶算子导致的双重困难，得到了正规化解的存在性。最后，我们研究了广义拟线性 Schr?dinger 方程的局域解（Localized solutions）。我们假设非线性项具有较弱的增长和单调性条件，利用罚函数和广义 Nehari 流形方法，得到了解的多重性和集中性。

Elliptic equations have extensive applications in various fields such as condensed matter physics, nonlinear optics, fluid mechanics, elasticity, finance and nonlinear diffusions processes. Research on relevant issues concerning solutions of elliptic equations has been one of the hotspots in the field of nonlinear partial differential equations, and many mathematicians have made a series of remarkable work. In this thesis, we study the existence and related properties of solutions for some important elliptic equations via variational methods and elliptic equations theories. Firstly, we study normalized solutions of the nonlinear elliptic system with Sobolev critical exponent, where the corresponding scalar equation has two nonlinear terms. In the mass supercritical case, we use a new technique to prove that the ground state energy of the scalar equation is monotonically non-increasing with respect to mass, and by combining the concentration compactness principle and fine estimates, we overcome the lack of compactness caused by the critical exponent, and then obtain the existence of the normalized solution. This is the first result for this system in the 3 or 4-dimensional case. In the mixed case, we decompose the Pohozaev manifold and prove that the energy functional has a local minimum point, which is also the ground state solution. Furthermore, by using contradiction methods, combining Hadamard’s three circles theorem and eigenvalue theory, we obtain some non-existence results. Secondly, we study normalized solutions of a class of fractional elliptic equations with a potential. By variational methods, we transform the existence of normalized solutions to the convergence of the minimizing sequences of the corresponding functional to the equations. Based on iterative techniques, we establish a new proof of sub-additive inequalities for the ?2-constraint minimization problem, which overcomes the double difficulties caused by the potential term and the fractional operator with non-local characteristic, and then we obtain the existence of the normalized solution. Finally, we consider localized solutions of the generalized quasilinear Schr?dinger equation. We assume that the nonlinear term has weaker growth and monotonicity conditions. By using the penalization method and the generalized Nehari manifold, we prove the multiplicity and concentration behavior of solutions to the equation.