Three-dimensional nonlinear problems with lack of compactness and mixed boundary conditions

We are interested in critical nonlinear problems involving the Laplacian operator with mixed Dirichlet-Neumann boundary conditions on bounded domains of $ \mathbb{R}^3 $. Using asymptotic analysis and further developments of the theory of critical points at infinity [7], we study the lack of compactness of the associated variational structures. Unlike previous works on related critical nonlinear problems, the present study does not impose any conditions on the critical points of the prescribed function. As a product, we prove a perturbation theorem through a new Euler-Hopf type criterion.