近年来，关于非线性椭圆型问题解的存在性、多解性与解的相关性质一直是广大研究者关注的问题，特别是关于分数阶椭圆型问题的研究. 在本项目中，我们将讨论几类分数阶椭圆型问题解的集中现象. 首先，我们考虑分数阶Schrödinger方程具有高维集中现象解的存在性；其次，我们将探讨分数阶临界方程“冠”形变号解的非退化性，利用此性质来研究分数阶渐近临界问题、分数阶Coron问题与带有非齐次项的分数阶临界问题的一类新的变号解与bubble-tower解的存在性；然后，我们讨论分数阶渐近超临界椭圆问题与分数阶Brezis-Nirenberg超临界问题具有高维集中现象解的存在性. 本项目主要涉及到非线性泛函分析、几何分析与偏微分方程等数学分支.

In recent years, the existence, multiplicity and properties of solutions for nonlinear elliptic problems has been attracted a lot of attention by many researchers, especially for fractional elliptic problems. This project is devoted to consider the concentration phenomenon of solutions for some fractional elliptic problems. Firstly, we consider the concentration at sub-manifolds of solutions for fractional Shcrödinger equation. Secondly, we will discuss the non-degeneracy of crown-shaped sign changing solutions of fractional critical equation. By using this property, we will study the existence of a new type sign-changing solution and bubble-tower solution for some fractional elliptic problems, such as the fractional asymptotic critical problem, fractional Coron problem and fractional critical problem with a nonhomogeneous term. Lastly, we study the high dimensional concentration phenomenon of solutions for fractional asymptotically supercritical elliptic problem and fractional Brezis-Nirenberg supercritical problem. This project mainly involves nonlinear functional analysis, geometric analysis and partial differential equations.