  To address the uniqueness issues associated with the Dirichlet problem for
the $N$-harmonic equation on the unit disk $\D$ in the plane, we investigate
the $L^p$ integrability of $N$-harmonic functions with respect to the standard
weights $(1-|z|^2)^{\alpha}$. The question at hand is the following. If $u$
solves $\Delta^N u=0$ in $\D$, where $\Delta$ stands for the Laplacian, and
[\int_\D|u(z)|^p (1-|z|^2)^{\alpha}\diff A(z)<+\infty,] must then
$u(z)\equiv0$? Here, $N$ is a positive integer, $\alpha$ is real, and
$0<p<+\infty$; $\diff A$ is the usual area element. The answer will, generally
speaking, depend on the triple $(N,p,\alpha)$. The most interesting case is
$0<p<1$. For a given $N$, we find an explicit critical curve
$p\mapsto\beta(N,p)$ -- a piecewise affine function -- such that for
$\alpha>\beta(N,p)$ there exist non-trivial functions $u$ with $\Delta^N u=0$
of the given integrability, while for $\alpha\le\beta(N,p)$, only $u(z)\equiv0$
is possible. We also investigate the obstruction to uniqueness for the
Dirichlet problem, that is, we study the structure of the functions in
$\mathrm{PH}^p_{N,\alpha}(\D)$ when this space is nontrivial. We find a
fascinating structural decomposition of the polyharmonic functions -- the
cellular (Almansi) expansion -- which decomposes the polyharmonic weighted
$L^p$ in a canonical fashion. Corresponding to the cellular expansion is a
tiling of part of the $(p,\alpha)$ plane into cells. A particularly interesting
collection of cells form the entangled region.
