Analysis and applications of nonlinear problems with lack of compactness

该研究项目涉及非线性偏微分方程和与非线性分析、黎曼几何和数学物理中的各种问题相关的系统的研究。我们提出研究的主要问题来自能量空间中方程的解（或近似解）可能缺乏紧性。紧致性的缺乏通常会产生浓度现象，这种现象以穗状溶液族（也称为爆破溶液）的形式出现。例如，当我们观察数学物理问题时，这些集中现象可以被视为缺乏稳定性。这些现象出现在纯数学和应用数学的几个重要方程中。我们将在这个程序中考虑的方程包括，除其他外，保形几何中的曲率处方方程，数学物理中的非线性Schrödinger-type方程和系统（更具体地说，在非线性光学和玻色-爱因斯坦凝聚体的Hartree-Fock理论中）以及非线性分析中sobolev型不等式的极值函数的欧拉-拉格朗日方程。在所有这些问题中，分析解决方案或近似解决方案的潜在缺乏紧性将是基本的。在某些情况下，我们将能够证明所有解族都是紧致的，即不存在任何爆破解族。这通常需要对解决方案进行非常精细的逐点分析。相反，在其他情况下，我们将能够找到爆破解族的存在。在这种情况下，出现了关于这些解决方案的概况和它们集中的点的位置的其他几个问题。这些问题的答案通常是一方面使用点式分析，另一方面使用基于所谓Lyapunov-Schmidt约简方法的构造方法。这些不同的案例、问题和方法都将在本节目中进行调查。我们还将通过使用变分方法来探讨解的存在性和多重性问题，在这里，紧致性的分析将再次发挥关键作用。总之，这个项目的目标是通过研究纯数学和应用数学不同领域的方程和系统，探索新型解决方案（例如我的提案第1节中讨论的符号变化和高能解决方案），具有复杂结构的系统（参见第2节），影响缺乏紧凑性问题的许多方面，以及具有更复杂算子的方程（如第3节中的高阶算子和第4节中的拟线性算子）。该计划的每个领域都将通过招聘几名HQP（博士后和各级学生）得到关键支持。

This research program is concerned with the study of nonlinear partial differential equations and systems connected to various problems in Nonlinear Analysis, Riemannian Geometry and Mathematical Physics. The main issue that we propose to investigate comes from potential lack of compactness of solutions (or approximating solutions) of the equations in energy spaces. The lack of compactness usually generates concentration phenomena, which take the form of families of spike solutions (also called blowing-up solutions). These concentration phenomena can be seen for example as a lack of stability when looking at problems from Mathematical Physics. These phenomena arise in several important equations in pure and applied Mathematics. The equations that we will consider in this program include, among others, curvature prescription equations in Conformal Geometry, nonlinear Schrödinger-type equations and systems in Mathematical Physics (more specifically, in nonlinear optics and the Hartree-Fock theory for Bose-Einstein condensates) and Euler-Lagrange equations of extremal functions to Sobolev-type inequalities in Nonlinear Analysis. In all these problems, analyzing the potential lack of compactness of solutions or approximating solutions will be fundamental. In some cases, we will be able to prove that all families of solutions are compact, namely that there does not exist any families of blowing-up solutions. This usually requires performing a very fine pointwise analysis of solutions. In other cases, on the contrary, we will be able to find existence of families of blowing-up solutions. In such cases, several other questions emerge about the profiles of such solutions and the locations of the points where they concentrate. These questions are usually answered by using pointwise analysis on the one hand and constructive methods based on the so-called Lyapunov-Schmidt reduction method on the other hand. These different cases, questions and methods will all be investigated in this program. We will also explore questions of existence and multiplicity of solutions by using variational methods, where, here again, the analysis of compactness will play a crucial role. In summary, this program will aim to impact the many aspects of problems with lack of compactness, by investigating equations and systems from different areas of pure and applied mathematics and exploring new types of solutions (such as for example sign-changing and high-energy solutions as discussed in Section 1 of my proposal), systems with complex structures (see Section 2), and equations with more complex operators (such as the higher-order operators in Section 3 and the quasilinear operators in Section 4). Each area of the program will be crucially supported by the recruitment of several HQP (postdoctoral fellows and students of every level).