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。

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.