Existence and non-existence of blowing-up solutions for nonlinear elliptic equations arising in physics and geometry

我将在这个研究项目中探索的数学取自数学物理、几何和非线性分析中的各种问题。为了理解这些问题，需要研究非线性偏微分方程组。在这个程序中，我将重点介绍椭圆型方程。我的主要目标是研究集中现象，即尖峰解(也称为爆破解)族的存在。在数学物理中，这种现象可以解释为方程缺乏稳健性。在几何学中，当研究度量的共形变形时，通常会出现这些问题。集中现象也出现在非线性分析的许多方面，例如研究Sobolev不等式的极值函数。在观察这类现象时，出现了几个有趣的问题。我们能确定在哪些情况下会发生这种情况吗？我们能描述存在这些情况的爆破解决方案的概况吗？我们能找到爆炸溶液可能的集中点吗？这些解的能量表现如何等。为了回答这些问题，我们有几种方法可供我们使用，包括逐点或能量空间中的爆破分析，以及构造方法，如Lyapunov-Schmidt约化。 基于我在爆破分析和构造方法方面的经验，我将在本节目中研究在各种情况下证明爆破解存在或不存在的新结果。我将在这个程序中讨论的主题将包括耦合的非线性薛定谔方程的稳定性和不稳定性，多峰解的存在性的研究，在Lyapunov-Schmidt约化过程中产生的退化问题，以及具有更复杂结构的方程的爆破分析，如各向异性方程，高阶方程，和完全非线性方程。

The mathematics I will explore in this research program are taken from various problems in mathematical physics, geometry, and nonlinear analysis. To understand these problems, study of nonlinear partial differential equations is required. In this program, I will focus on equations of elliptic type. My main goal is to study concentration phenomena, namely the presence of families of spike solutions (also called blowing-up solutions). In mathematical physics, such phenomena can be interpreted as a lack of robustness of the equations. In geometry, these typically arise when studying conformal deformations of metrics. Concentration phenomena also appear in many aspects of nonlinear analysis such as for instance the study of extremal functions of Sobolev inequalities. Several interesting questions arise when looking at this type of phenomena. Can we identify in which situations this occurs? Can we describe the profiles of blowing-up solutions where these exist? Can we locate the possible concentration points of blowing-up solutions? How does the energy of such solutions behave, etc. To answer these questions, we have several types of methods at our disposal including blow-up analysis, either pointwise or in energy spaces, and constructive methods such as the Lyapunov-Schmidt reduction. Based on my experience of both blow-up analysis and constructive methods, I will in this program investigate new results which show the existence or nonexistence of blowing-up solutions in various contexts. Topics which I will address in this program will include the stability and instability of coupled nonlinear Schrödinger equations, an investigation into the existence of multi-spike solutions, problems of degeneracy arising in the Lyapunov-Schmidt reduction, and blow-up analysis for equations with more complex structures such as anisotropic equations, higher-order equations, and fully nonlinear equations.