Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry

Abstract

Positive entire solutions of the singular biharmonic equation $Delta^2 u + u^{-q}=0$ in $mathbb{R}^n$ with $q>1$ and $ngeq 3$ are considered. We prove that there are infinitely many radial entire solutions with different growth rates close to quadratic. If $u(0)$ is kept fixed we show that a unique minimal entire solution exists, which separates the entire solutions from those with compact support. For the special case $n=3$ and $q=7$ the function $U(r) = sqrt{1/sqrt{15}+r^2}$ is the minimal entire solution if $u(0)=15^{-1/4}$ is kept fixed.

Keywords

• Singular biharmonic equation
• conformal invariance.