  We study existence and stability properties of entire solutions of a
polyharmonic equation with an exponential nonlinearity. We study existence of
radial entire solutions and we provide some asymptotic estimates on their
behavior at infinity. As a first result on stability we prove that stable
solutions (not necessarily radial) in dimensions lower than the conformal one
never exist. On the other hand, we prove that radial entire solutions which are
stable outside a compact always exist both in high and low dimensions. In order
to prove stability of solutions outside a compact set we prove some new
Hardy-Rellich type inequalities in low dimensions.
