At the CEREMADE I would like to develop a research program concerning the study of some classes of differential equations through variational methods, having as common feature a lack of compactness.
1. Homoclinic and heteroclinic orbits for Hamiltonian systems. This problem has been recently investigated with variational techniques starting with some works by V. Coti Zelati, I. Ekeland, E. Sere and P. Rabinowitz who considered the autonomous and time periodic cases. I would like to investigate the case of general oscillating time dependence. 2. Positive solutions for semilinear elliptic equations in unbounded domains. This subject is related to the first one and the techniques used to tackle the above problems could be suitably modified to get existence results also in this case.
3. Stationary states of a class of nonlinear relativistic wave equations. This problem has been recently studied by E. Sere and M.J. Esteban with some methods inspired by those used for the above problems. I would like to study these new techniques and to consider some open questions.