研究多重的Lane-Emden方程，给出有限Morse指标解的完全分类。由于问题的特殊性，计划解决重数大于5的情形（不同重数之间有极大的差别)；对于分数阶（非局部）Lane-Emden方程，解决该类方程的有限Morse指标解的完全分类。利用Gamma函数，确定Joseph-Lundgren 指数的精确公式。关键的科学问题是建立新的单调性公式。研究“上临界”的Choquard方程, 利用变分法和非局部补偿技巧，寻找位势的充分条件，证明方程存在一族半经典解、且集中到位势的局部极小值点；并且确保参数的范围和位势的衰减假设是最优的； 解决该临界方程正规化解的存在性问题。在Chern-Simons 模型方面，对于Rank=2, 3或者更高时，我们将寻找充分条件，建立非拓扑解的存在性定理；这些条件取决于奇异sources以及它的个数。拟采用的办法是奇异sources的扰动、变分与拓扑方法。

We classify the s finite Morse index solutions of the arbitrary polyharmonic Lane-Emden equation by new methods. We provide a new perspective and a deep understanding to classify the homogeneous solutions and construct the coercive estimates in large dimensions hence classify the stable and finite Morse index solutions in large dimensions. We also derive a novel monotonicity formula and obtain a Liouville theorem of the nonlocal Lane-Emden equation. By using Gamma function, we give the precise formula for the Joseph-Lundgren exponent. The fundamental method is the novel monotonicity formula. We study the the nonlinear Choquard equations with upper-critical exponent. We show the existence of solutions of the equations by introducing some new sufficient conditions..We.find the normalized solutions for a class of nonlinear Choquard equations with a prescibed norm. On the Chern-Simons model, we consider the existence and uniqueness of the non-topological solutions. We also study the existence and uniqueness of the mixed type solutions. Further, we will study the new type of nontopological bubbling solutions in the Chern–Simons model in the case of Rank>2. We will seek the sufficient conditions which may produce non-topological solutions. These conditions may be depending on the number of the singularsources. The perturbation of singular sources and the variational and topological methods will be involved.