The singular extremal solutions of the bi-Laplacian with exponential nonlinearity

Consider the problem

$\left\{ \begin{array}{ll} \Delta^2 u= \lambda e^{u} &\text{in } B, u=\frac{\partial u}{\partial n}=0 &\text{on }\partial B, \end{array} \right.$

where $B$ is the unit ball in ${\mathbb{R}}^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln(\vert x\vert)$, 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. Recently a computer-assisted proof was used to show that the extremal solution is singular in dimensions $13\leq N\leq 31$. Here by an improved Hardy-Rellich inequality we overcome this difficulty and give a simple mathematical proof to show that the extremal solution is singular in dimensions $N\geq13$.