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

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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.