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