Contributions to Nonlinear PDEs arising in Conformal Geometry, Mean Field Games and Choquard models.

The goal of this thesis is to analyze certain nonlinear elliptic Partial Differential Equations (briefly PDEs) that arise in Conformal Geometry, Mean Field Games theory, and Choquard models. Our primary focus is on the study of the existence and nonexistence of solutions, analyzing their asymptotic behavior, and examining the occurrence of concentrating phenomena. The contents of this manuscript have been organized into three distinct parts, each one focusing on a specific topic. The first part of this work deals with some prescribed curvature problems. In particular, in Chapter 1 we address the problem of existence and compactness of entire solutions to the Gaussian curvature equation on R^2 in the case of power-type and sign-changing prescribed curvature; while in Chapter 2 we deal with the corresponding prescribed Q-curvature problem in R^4. While these two issues are undoubtedly interconnected, it is essential to recognize that each possesses distinct features that warrant in-depth analysis and attention. The content of Part I corresponds to the research papers [17, 18]. In the second part, we study second-order ergodic Mean-Field Games systems defined in the whole space R^N with a coercive potential V and aggregating nonlocal coupling, given in terms of a Riesz-type interaction kernel. In Chapter 3, we prove that the strength of the attractive term and the behavior of the diffusive part interact to produce three distinct regimes for the existence and nonexistence of classical solutions in our MFG system. On the other hand, in Chapter 4, exploiting a variational approach and a concentration-compactness argument, we show that in the vanishing viscosity limit, there is concentration of mass around minima of the potential V . This leads to proving the existence of solutions to the potential-free system. The content of Part II corresponds to the research papers [19, 20]. The third part of this thesis focuses on boundary value problems for Choquard equations. More in detail, we investigate the existence of solutions when the domain is an annulus or an exterior domain, considering both Neumann and Dirichlet boundary conditions. The results of Chapter 5 are presented in the work [21].