Classification of the stable solution to biharmonic problems in large dimensions

TL;DR

This paper establishes a new bound on the exponent for the nonexistence of stable solutions to a biharmonic problem in high dimensions, advancing understanding of solution stability in nonlinear PDEs.

Contribution

It provides a novel bound on the exponent p for which stable solutions do not exist in large dimensions, specifically for n ≥ 20.

Findings

New bound on the exponent p for nonexistence of stable solutions

Applicable to dimensions n ≥ 20

Enhances understanding of stability in biharmonic equations

Abstract

We give a new bound on the exponent for the nonexistence of stable solutions to the biharmonic problem $$\Delta^2u=u^p,\quad u>0 in \mathbb{R}^n $$ where $p>1, n \geq 20$.