分数阶非线性椭圆方程多峰解的若干研究

In this project, we discuss three fractional elliptic problems: the first is fractional Schrödinger equations and fractional field equations on whole spaces, the second is fractional Dirichlet and fractional Neumann problems with critical exponents, and the third is fractional Bose-Einstein systems. We will use Lyapunov-Schmidt reduction method to study multipeak solutions and infinite many solutions for the above three problems. It is well-known that fractional Laplace operator is nonlocal and Laplace operator is local. Moreover, there is a great difference between the decay rate of solutions to the corresponding limit equations for fractional Laplace operator and Laplace operator. Therefore, by studying multipeak solutions for fractional elliptic problems, it is useful for us to understand commonness and difference between fractional elliptic problems and classical elliptic problems.

本项目拟讨论三类分数阶椭圆问题：第一类是全空间分数阶Schrödinger方程和场方程；第二类是含临界指标的分数阶Dirichlet问题和Neumann问题；第三类是分数阶Bose-Einstein方程组。我们拟采用Lyapunov-Schmidt约化方法研究这三类问题的多峰解与无穷多解。众所周知，分数阶Laplace算子是非局部的，而Laplace算子是局部的，而且它们对应的极限方程的解在衰减速度上也有很大的区别。因此，对分数阶椭圆问题多峰解的研究，有利于我们进一步揭示分数阶椭圆问题与经典的椭圆问题之间的共性与区别。

近年来，非局部分数阶椭圆问题的研究引起了国内外数学界的广泛关注，其研究十分活跃，意义深远。本项目对分数阶椭圆型方程的多峰解和多解问题进行研究。具体来说：我们讨论了全空间分数阶椭圆问题无穷多解的存在性和集中性；研究了含临界指标的分数阶椭圆问题；还探讨了含其他非局部项椭圆问题的多解存在性。通过这些问题的研究，我们在一定程度上揭示了分数阶椭圆问题与经典椭圆问题的共性与区别。