  Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive
integer let $ \Lambda_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3,
$ be a $k$-dimensional plane with $ 0 \in \Lambda_k.$ If $n-k < p <\infty$, we
first study the Martin boundary problem for solutions to the $p$-Laplace
equation (called $p$-harmonic functions) in $ \mathbb{R}^{n} \setminus
\Lambda_k $ relative to $ \{0\}. $ We then use the results from our study to
extend the work of Wolff on the failure of Fatou type theorems for $p$-harmonic
functions in $ \mathbb{R}^{2}_+ $ to $p$-harmonic functions in $ \mathbb{R}^{n}
\setminus \Lambda_k $ when $ n-k < p <\infty$. Finally, we discuss
generalizations of our work to solutions of $ p $-Laplace type PDE (called $
\mathcal{A}$-harmonic functions).
