Blow-up and clustering configurations: new solutions to the Brézis-Nirenberg Problem

This thesis investigates a nonlinear elliptic partial differential equation on bounded domains, involving the critical Sobolev exponent: the well-known Brézis-Nirenberg problem with Dirichlet boundary conditions. We focus on establishing new types of solutions that deepen our understanding of blow-up and clustering phenomena within this framework. Our findings include the existence of positive solutions in dimension 4 that exhibit blow-up at a single point of the domain in the non-autonomous case. Additionally, we present a novel type of sign-changing solution that clusters at a single boundary point. While such clustering configurations have been observed in other equations, this is the first result of its kind for the Brézis-Nirenberg problem.