Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

TL;DR

This paper investigates the extremal solution of a supercritical biharmonic equation with power nonlinearity, proving its uniqueness and singularity in high dimensions for large exponents.

Contribution

It establishes the uniqueness of the extremal solution and characterizes its singularity for high dimensions and large nonlinear exponents.

Findings

The extremal solution is unique for the given problem.

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

The solution's behavior is bounded by a specific power-law near the origin.

Abstract

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has a solution, where $\B$ is the unit ball in $R^{n}$ centered at the origin, $p>\frac{n+4}{n-4}$ 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, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the unit ball.