Existence of stable solutions to $(-\Delta)^m u=e^u$ in $\mathbb{R}^N$ with $m \geq 3$ and $N > 2m$

Abstract

We consider the polyharmonic equation $(-\Delta)^m u=e^u$ in $\mathbb{R}^N$ with $m \geq 3$ and $N > 2m$. We prove the existence of many entire stable solutions. This answer some questions raised by Farina and Ferrero.