  This paper is concerned with the $L^p$ integrability of $N$-harmonic
functions with respect to the standard weights $(1-|x|^2)^{\alpha}$ on the unit
ball $\mathbb{B}$ of $\mathbb{R}^n$, $n\geq 2$. More precisely, our goal is to
determine the real (negative) parameters $\alpha$, for which
$(1-|x|^2)^{\alpha/p} u(x) \in L^p(\mathbb{B})$ implies that $u\equiv 0$,
whenever $u$ is a solution of the $N$-Laplace equation on $\mathbb{B}$. This
question is motivated by the uniqueness considerations of the Dirichlet problem
for the $N$-Laplacian $\Delta^N$.
  Our study is inspired by a recent work of Borichev and Hedenmalm [Adv. Math.,
264(2014), pp. 464-505], where a complete answer to the above question in the
case $n=2$ is given for the full scale $0<p<\infty$. When $n\geq 3$, we obtain
an analogous characterization for $\frac{n-2}{n-1}\leq p<\infty$, and remark
that the remaining case can be genuinely more difficult. Also, we extend the
remarkable cellular decomposition theorem of Borichev and Hedenmalm to all
dimensions.

Complex Variables;Analysis of PDEs