Nonlinear elliptic systems with critical growth

In the first part we study a class of semi-linear and quasi-linear systems which describe the interaction between charged fields or quantum particles with an unknown external gauge. We consider the limiting case when the nonlinearity exhibits the maximal growth which allows us to treat such problems by a variational approach. In particular we obtain non-existence results by Pohozaev-type techniques and then, adding lower-order perturbations we recover existence of mountain pass type solutions. In the second part, we consider semi-linear Hamiltonian systems where the nonlinearities enjoy a suitable coupled critical growth and in bounded domains. We provide a direct variational approach to construct a min-max level of linking type, involving explicit concentrating directions which affect the problem with a lack of compactness. Finally, we look at the systems considered in the second part but with inequalities instead of equalities and in the whole space. Exploiting non-linear capacity arguments introduced by S. Pohozaev we give a new proof and a generalization of a non-existence result by E.Mitidieri. In particular we show that the system of differential inequalities has no nontrivial solutions, in an optimal weak sense, for the nonlinearities satisfying a sharp growth condition. Here the case of dimension two is also covered.