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是从数学物理，几何和非线性分析中的各种问题中提取的。为了理解这些问题，需要研究非线性偏微分方程。在这个程序中，我将集中在椭圆型方程。我的主要目标是研究浓度现象，即存在的家庭的尖峰解决方案（也称为爆破解决方案）。在数学物理学中，这种现象可以解释为方程缺乏鲁棒性。在几何学中，当研究度量的共形变形时，这些通常会出现。集中现象也出现在非线性分析的许多方面，例如Sobolev不等式极值函数的研究。在观察这类现象时，会出现几个有趣的问题。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解的爆破解的轮廓吗？我们能找到爆炸溶液的可能的浓度点吗？如何这样的解决方案的能量行为，等等，为了回答这些问题，我们有几种类型的方法在我们的处置，包括爆破分析，无论是逐点或在能量空间，和建设性的方法，如李雅普诺夫-施密特减少。 根据我的爆破分析和建设性的方法的经验，我将在这个程序中调查新的结果，显示存在或不存在的爆破解决方案在各种情况下。我将在这个程序中解决的主题将包括耦合非线性薛定谔方程的稳定性和不稳定性，调查多尖峰解的存在性，在Lyapunov-Schmidt还原中产生的退化问题，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.