Existence and stability properties of entire solutions to the polyharmonic equation $(-\mathrm{\Delta }{)}^{m}u=e^u$ for any $m\ge 1$

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.