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On stable solutions of biharmonic problem with polynomial growth

We prove the nonexistence of smooth stable solution to the biharmonic problem $Δ^2 u= u^p$, $u>0$ 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 $Δ^2 u= ł(u+1)^p$ 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.