Analysis and applications of nonlinear problems with lack of compactness

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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).