Properties of the extremal solution for a fourth-order elliptic problem

TL;DR

This paper investigates the extremal solution of a fourth-order elliptic problem in a unit ball, proving its uniqueness at the extremal parameter and analyzing its singularity for high dimensions and large p.

Contribution

It establishes the existence, uniqueness, and singularity properties of the extremal solution for a class of fourth-order elliptic equations, extending understanding of solution behavior in high dimensions.

Findings

The extremal solution is unique at the critical parameter.

The extremal solution is singular when the dimension is at least 13 for large p.

The solution bounds are explicitly characterized within the unit ball.

Abstract

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\frac{\lambda}{(1-u)^{p}} & \{in}\ \ B, 0<u\leq 1 & \{in}\ \ B, u=\frac{\partial u}{\partial n} =0 & \{on}\ \ \partial B. {array} . $$ has a solution, where $B$ is the unit ball in $R^{n}$ centered at the origin, $p>1$ and $n$ is the exterior unit normal vector. We show that for $\lambda=\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough and $1-C_{0}r^{\frac{4}{p+1}}\leq u^{*}(x)\leq 1-r^{\frac{4}{p+1}}$ on the unit ball, where $ C_{0}:=(\lambda^{*}/\bar{\lambda})^{\frac{1}{p+1}}$ and $\bar{\lambda}:=\frac{8(p-1)}{(p+1)^{2}}[n-\frac{2(p-1)}{p+1}][n-\frac{4p}{p+1}]$. Our results actually complete part of the open problem which \cite{D} lef