In this talk, will be presented some results on the optimal constants of Sobolev and Hardy-Sobolev inequalities with weights and their relation with the behavior of some mixed Dirichlet-Neumann boundary conditions. Will be analyzed the attainability of the Sobolev constant and the Hardy-Sobolev constant. The deep relation between the geometry of the domain, the boundary conditions and the attainability of the critical constants will be showed. As a direct application some applications to elliptic problems will be discussed.