Nonlinear variational problems with lack of compactness.
In this thesis we deal with two different classes of variational problems: 1) the problem of closed curves with prescribed curvature, or $H$-loop problem; 2) the study of the nodal solutions of the fractional Brezis-Nirenberg problem. In both cases we deal with nonlinear equations (an ODE system for problem 1, and an elliptic equation for problem 2) which admit a variational structure. Nevertheless, in both cases, a lack of compactness occurs and this constitutes a strong obstruction for the application of the direct methods of the calculus of variations or even standard variational methods. Therefore, we are lead to use more refined techniques or also to follow different approaches, like the Lyapunov-Schmidt method and blow-up analysis. This will allow us not only to obtain existence and multiplicity results, but also qualitative properties of the solutions.