Towards a Proof of Bahri-Coron's Type Theorem for Mixed Boundary Value Problems

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of R-n,n = 3 and mixed Dirichlet-Neumann boundary conditions. We study the effect of the domain's topology on the existence of solutions as Bahri-Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri-Coron and demands new constructions and extra ideas.