全空间上非局部椭圆型方程解的存在性及其渐近行为的研究

In this project, we study three types of nonlocal elliptic equations. By using variational methods, we deal with the existence, multiplicity and asymptotic behavior of solutions. These equations are mathematical models which come from Physics, Chemistry, Materials science, Biology etc., and they have profound applied background. However, it is difficult to solve these problems due to the presence of nonlocal term and the lack of compactness. We shall do our best to overcome these two difficulties in this project. At first, we study the existence, multiplicity and asymptotic behavior of solutions for planar Schrödinger-Poisson systems. Then, we consider the existence and other related problems of normalized solutions for fractional Schrödinger equations, and we focus on the blow-up behavior of normalized solutions in the mass critical case. Finally, we deeply discuss the existence and properties of solutions for Kirchhoff problems with steep potential well. In particular, we are concerned with the existence and asymptotic behavior of ground state solutions for Kirchhoff problems with critical growth. By means of the research of this project, we hope to make some contributions to the development of nonlinear analysis in nonlocal elliptic equations.

本项目将以三类非局部椭圆型方程为研究对象，利用变分方法研究解的存在性、多解性以及解的渐近行为。这些方程是物理学、化学、材料科学及生物学等诸多领域中一些问题的数学模型，有着十分丰富的应用背景。但由于存在非局部项和缺乏紧性，这些问题的研究有很大难度，本项目力求对上述两个难点有所突破。首先，研究二维薛定谔－泊松系统解的存在性与多重性，以及解的渐近行为；然后，考虑分数阶薛定谔方程规范解的存在性及其相关问题，重点关注在质量临界情形下规范解的爆破性；最后，对带陡峭势阱的基尔霍夫问题解的存在性及其性质进行深入探讨，特别关注在临界指数增长情形下基态解的存在性及其渐近行为。我们期望通过本课题的研究，推进非线性分析理论在非局部椭圆型方程中的发展。

本项目以非局部椭圆型方程为研究对象，利用变分方法研究解的存在性、多解性以及解的渐近行为。这些方程是物理学、化学、材料科学及生物学等诸多领域中一些问题的数学模型，有着十分丰富的应用背景。但由于存在非局部项和缺乏紧性，这些问题的研究有很大难度。本项目获得的主要结果有：首先，我们研究了带陡峭势阱的基尔霍夫问题，证明了正解的存在性及其渐近行为，并给出了正解在无穷远处的衰减率；其次，我们考虑了带有临界指数增长的拟线性薛定谔方程，证明了正解的多重性与集中性；再次，我们研究了带囚阱势的双耦合哈里特方程组，证明了基态解的存在性和不存在性，以及基态解的极限行为；最后，我们考虑了带陡峭势阱的三维薛定谔－泊松系统，证明了正解的存在性和不存在性，以及正解的渐近行为。