The singular extremal solutions of the bilaplacian with exponential nonlinearity

TL;DR

This paper proves that the extremal solutions of a biharmonic equation with exponential nonlinearity are singular in dimensions 13 and higher, using improved Hardy-Rellich inequalities to provide a rigorous mathematical proof.

Contribution

It offers a new, simpler mathematical proof for the singularity of extremal solutions in high dimensions, improving upon previous computer-assisted methods.

Findings

Extremal solutions are singular for dimensions N ≥ 13.

Improved Hardy-Rellich inequality is key to the proof.

Simplifies the understanding of solution behavior in high dimensions.

Abstract

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 $\R^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln(|x|)$, 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 $13\leq N\leq 31$. 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 $N\geq13$.