Barycenter Technique and Bahri-Brezis-Nirenberg type problems

Abstract

In this dissertation, we study three Bahri-Brezis-Nirenberg type problems related to the classical Yamabe problem from conformal geometry by using the Barycenter Technique of Bahri-Coron[8]. Many problems in Mathematics, Physics, and the Natural Sciences can be modeled using semilinear elliptic boundary value problem with non-linearity critical with respect to some Sobolev type inequality. In the first problem, we discuss the Brezis-Nirenberg problem on bounded smooth domains of R^3. Using the celebrated Algebraic Topological argument (also called Barycenter Technique) of Bahri-Coron[8] as implemented in [23] combined with the Brendle[10] Schoen[30]’s bubble construction, we provided a solution for non-contractible domains under the assumption that the involved operator has a positive first eigenvalue and a positive Green’s function. In the second problem, we discuss a Cherrier-Escobar problem for the extended problem corresponding to the elliptic Schr¨odinger-to Neumann map on a compact 3-dimensional Riemannian manifold with boundary. Using the Algebraic Topological argument of Bahri-Coron[8], we showed solvability under the assumption that the extended problem corresponding to the elliptic Schr¨odinger-to-Neumann map has a positive first eigenvalue and a positive Green’s function. In the third problem, we discuss a Bahri-Brezis type problem on a compact 3-dimensional asymptotically hyperbolic manifold. Using the Algebraic Topological argument of Bahri-Coron[8], we showed the existence of at least one solution under the assumption that the corresponding degenerate boundary value problem has a positive first eigenvalue and a positive Green’s function.