  We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $
\mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B(
0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 <
\infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic
function, $ V = V ( r e^{i\theta} )$, which is $ 2\pi/N $ periodic in the $
\theta $ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with
Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $
c^{-1} \leq \int_{-\pi}^{\pi} V ( e^{i\theta} ) d \theta \leq c$. In case
$2<p<\infty $ we give a more or less explicit example of $V$ and our work is an
extension of a result of Wolff on $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$. Using
our first result, we extend the work of Wolff on failure of Fatou type theorems
for $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$ for $p$-harmonic functions, $1< p\neq
2<\infty$. Finally, we also outline the modifications needed for extending the
work of Llorente, Manfredi, and Wu regarding failure of subadditivity of
$p$-harmonic measure on $ \partial \mathbb{R}^{2}_+ $ to $\partial B (0, 1)$.
