  In this paper we develop a Gidas-Ni-Nirenberg technique for polyharmonic
equations and systems of Lane-Emden type. As far as we are concerned with
Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth
order equations, namely which involve the fourth iteration of the Laplace
operator. Then, we can extend the result to arbitrary polyharmonic operators of
any order, provided some natural boundary conditions are satisfied but not for
Dirichlet's: the obstruction is apparently a new phenomenon and seems due to
some loss of information though far from being clear. When the polyharmonic
operator turns out to be a power of the Laplacian, and this is the case of
Navier's boundary conditions, as byproduct uniqueness of solutions holds in a
fairly general context. New existence results for systems are also established.
