Existence, non existence and uniqueness results for higher order elliptic systems

In this dissertation, we deal with Hamiltonian Lane-Emden type systems where in place of the Laplace operator we take into account the polyharmonic operator. Recallthatthepolyharmonicoperatordoesnotalwayssatisfyamaximumprinciple. Indeed, if we take Dirichlet boundary conditions, the maximum principle is known to hold only on a ball or small deformations of a ball, whereas it fails on ellipses with sufficiently big ratio of half axes. On the one hand, if the operators in the two equations have the same order, the problem is variational. In this case, we prove some existence and non-existence results under Dirichlet boundary conditions on a sufficiently smooth bounded domain. We also consider more general nonlinearities than power-like. The existence result is achieved by means of a Linking Theorem on suitable fractional Sobolev spaces. On the other hand, if the operators have different orders the problem turns out to be non-variational: here we prove some existence results on a ball by means of a different approach, precisely we exploit degree theory and a blow-up analysis, combined with a suitably adapted moving planes technique and Liouville-type theorems. Moreover, we prove uniqueness of solutions to equations and systems up to order eight on a ball endowed with Dirichlet boundary conditions. The proof extends the classical argument by Gidas-Ni-Nirenberg.