带有非局部项椭圆型方程变号解的存在性及其渐近行为

In this project, we will investigate the following three types of nonlinear elliptic equations: (1) Kirchhoff type equations which is related to the classical D'Alembert's wave equations; (2) a gauged nonlinear Schrodinger equation which is associated to the standing waves of Chern-Simons equations; (3) the fractional Laplacian equations. All of those equations come from either the physical models or the biological phenomena. Moreover, due to the appearance of the different nonlocal terms in the above three models, they are also called the nonlocal problems. Therefore, understanding the effect of the nonlocal terms will not only help us improving the mathematical analysis, but also help us interpretating the physical/biological applications. More precisely, we will consider the existence of least energy sign-changing solutions and multiplicity of sign-changing solutions to those three models, as well as the asymptotic behavior and the profile of those sign-changing solutions. The goal of this project is to further discover this exciting interplay between mathematics and physics/biology, and to introduce new techniques in partial differential equations, the calculus of variations, and nonlinear functional analysis which are both inspired by and shed new light upon these phenomena.

本项目我们主要研究下列三类非线性椭圆方程：（1）源于 D'Alembert 波方程的 Kirchhoff 型方程；（2）与 Chern-Simons 方程驻波解相关的带有校正场的薛定谔方程；（3）分数次 Laplacian 方程。这三类方程都起源物理模型或生物现象，而与经典的半线性方程相比，这三类方程中都含有非局部项，被称为非局部问题。因此全面清楚地认识非局部项的影响，不仅可以帮助我们改进数学分析方法，还可以帮助我们更好地诠释物理/生物应用。准确的说，我们主要研究这三类椭圆型方程极小能量变号解，多变号解的存在性，同时对解的渐近行为和其形态进行分析研究。这个项目的目标是进一步发现数学和物理/生物科学之间的相互作用关系，在偏微分方程，变分法和非线性泛函分析方面引入新的方法和技巧，从而能更好地揭示这些物理/生物现象的本质。

按照项目申请书中的研究计划，本项目主要研究了带非局部椭圆方程及其相关的几类变分问题，重点研究了下列三类非线性椭圆方程：（1）源于 D'Alembert 波方程的 Kirchhoff 型方程；（2）与 Chern-Simons 方程驻波解相关的带有校正场的薛定谔方程；（3）分数次 Laplacian 方程。我们主要研究这三类椭圆型方程极小能量变号解，多变号解的存在性，同时对解的渐近行为和其形态进行分析研究。整体上完成了项目拟定的研究目标，取得了一些有意义的研究进展，项目组在 Calc. Var. Partial Differential Equations、 J. Differential Equations、Nonlinearity等国际数学期刊上发表论文7篇。这些结果将帮助我们进一步发现数学和物理/生物科学之间的相互作用关系，在偏微分方程，变分法和非线性泛函分析方面引入新的方法和技巧，从而能更好地揭示这些物理/生物现象的本质。