On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta {\mathrm{\Delta }}^{2}u-\tau \mathrm{\Delta }u=\frac{\lambda }{(1-u{)}^2}$ on a ball $B \subset \R^N$,underNavierboundaryconditions$u=\Delta u=0$on$\partial B$, where $\lambda >0$ is a parameter, while $\tau >0$, $\beta >0$ are fixed constants. It is known that there exists a ${\lambda }^{\ast }$ such that for $\lambda >{\lambda }^{\ast }$ there is no solution while for $\lambda <{\lambda }^{\ast }$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{\ast }$ is regular (${\sup }_B{u}^{\ast }<1$) for $N\le 8$ and $\beta ,\tau >0$ and it is singular (${\sup }_B{u}^{\ast }=1$) for $N\ge 9$, $\beta >0$, and $\tau >0$ with $\frac{\tau }{\beta }$ small. Our proof for the singularity of extremal solutions in dimensions $N\ge 9$ is based on certain improved Hardy-Rellich inequalities.