Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded ?
If $\Phi$ is the fundamental solution of Laplace equation defined by $$\Phi(x)=\begin{cases}-\frac1{2\pi}\log|x|& \text{if }n=2\\\frac{1}{n(n-2)\alpha(n)|x|^{n-2}}&\text{if }n\ge 3\end{cases}$$ where $\alpha(n)$ is the volume of the unit ball in dimension $n$
Hint: Compute it directly using $n$-dimensional spherical coordinates. Note that $\phi(x)$ is only dependent on $|x|$. The computation gets quite easy, since for any $f(x)=f(|x|)$ we have $$\int_{B_\epsilon} f(|x|) dx = \int_0^\epsilon f(r) r^{n-1} dr \int_{\mathbb{S}^{n-1}} d\Omega.$$