The singular extremal solutions of the bilaplacian with exponential nonlinearity

Consider the problem {ll} \Delta^2 u= \lambda e^{u} &$\text{in}$ B u=\frac{\partial u}{\partial n}=0 &$\text{on}$\partial B, where $B$ is the unit ball in ${\mathbb{R}}^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln (\mathrm{\mid }x\mathrm{\mid })$, for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. D'avila et al. in \cite{DDGM} used a computer assisted proof to show that the extremal solution is singular in dimensions $13\le N\le 31$. Here by an improved Hardy-Rellich inequality which follows from the recent result of Ghoussoub-Moradifam \cite{GM} we overcome this difficulty and give a simple mathematical proof to show the extremal solution is singular in dimensions $N\ge 13$.