Abstract

We first obtain Liouville type results for stable entiresolutions of the biharmonic equation $-\Delta^2 u=u^{-p}$ in $\mathbb{R}^N$ for $p>1$ and $3 \leq N \leq 12$. Then weconsider the Navier boundary value problem for the correspondingequation and improve the known results on the regularity of theextremal solution for $3 \leq N \leq 12$. As a consequence, in the case of $p=2$, we show that the extremal solution $ u^{*}$ is regular when $N =7$. This improves earlier results of Guo-Wei [21] ($N \leq 4$),Cowan-Esposito-Ghoussoub [2] ($N=5$), Cowan-Ghoussoub [4] ($N=6$).

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