Growth properties of $p$-th means of biharmonic Green potentials in the unit ball

Abstract

Let $u$ be a biharmonic Green potential on the unit ball $\mathbf{B}$ of $\mathbf{R}^{n}$. We show that \begin{equation*} \lim_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0 \end{equation*} for $p$ such that $1\le p<(n-1)/(n-4)$ in case $n\ge 5$ and $1\le p<\infty$ in case $n\le 4$. Further, if $n\ge 5$ and $(n-1)/(n-4)\le p<(n-1)/(n-5)$, then it is shown that \begin{equation*} \liminf_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0. \end{equation*} Finally we show that these limits characterize biharmonic Green potentials among super-biharmonic functions on $\mathbf{B}$.