On stable solutions of biharmonic problem with polynomial growth

Abstract

We prove the nonexistence of smooth stable solution to the biharmonic problem ${\Delta }^{2}u=u^p$, $u>0$ in ${\mathbb{R}}^N$ 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 $N\le 12$ and $p>1$. Moreover, we consider also the corresponding problem in the half space ${\mathbb{R}}_{+}^N$, or the elliptic problem ${\Delta }^{2}u=ł(u+1{)}^p$ on a bounded smooth domain $Ø$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works.

Corrected typos

References in corpus (2)

• The Gel'fand problem for the biharmonic operator
• Supercritical biharmonic equations with power-type nonlinearity

Cited by in corpus (1)

• On weak solutions to a fractional Hardy-Hénon equation: Part II: Existence