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We study existence and stability properties of entire solutions of apolyharmonic equation with an exponential nonlinearity. We study existence ofradial entire solutions and we provide some asymptotic estimates on theirbehavior at infinity. As a first result on stability we prove that stablesolutions (not necessarily radial) in dimensions lower than the conformal onenever exist. On the other hand, we prove that radial entire solutions which arestable outside a compact always exist both in high and low dimensions. In orderto prove stability of solutions outside a compact set we prove some newHardy-Rellich type inequalities in low dimensions.

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