On stable solutions of biharmonic problem with polynomial growth

TL;DR

This paper establishes nonexistence of smooth stable solutions for certain biharmonic equations in Euclidean space and half-spaces, and proves regularity of extremal solutions in lower dimensions, advancing previous research in the field.

Contribution

It provides new nonexistence results for stable solutions of biharmonic problems with polynomial growth and improves understanding of extremal solutions' regularity.

Findings

Nonexistence of stable solutions for N < 2(1 + x_0)

Regularity of extremal solutions in lower dimensions

Extension of results to half-space and bounded domains

Abstract

We prove the nonexistence of smooth stable solution to the biharmonic problem $\Delta^2 u= u^p$, $u>0$ in $\R^N$ for $1 < p < \infty$ and $N < 2(1 + x_0)$, where $x_0$ is the largest root of the following equation: $$x^4 - \frac{32p(p+1)}{(p-1)^2}x^2 + \frac{32p(p+1)(p+3)}{(p-1)^3}x -\frac{64p(p+1)^2}{(p-1)^4} = 0.$$ In particular, as $x_0 > 5$ when $p > 1$, we obtain the nonexistence of smooth stable solution for any $N \leq 12$ and $p > 1$. Moreover, we consider also the corresponding problem in the half space $\R^N_+$, or the elliptic problem $\Delta^2 u= \l(u+1)^p$ on a bounded smooth domain $\O$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works.