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

TL;DR

This paper proves the existence of numerous stable solutions to the polyharmonic equation $(- riangle)^m u = e^u$ in high-dimensional Euclidean space for $m \\geq 3$, addressing open questions in the field.

Contribution

It establishes the existence of many entire stable solutions for the polyharmonic exponential equation in the case $m \\geq 3$, $N > 2m$, answering previously open questions.

Findings

Existence of many entire stable solutions.

Addresses open questions by Farina and Ferrero.

Results hold for $m \\geq 3$, $N > 2m$.

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.