We have the fundamental solution $\Phi(x)$ of Laplace's equation, $$ \Phi(x)=\left\{ \begin{array}{ll} -\frac{1}{2\pi}\log\lvert x\rvert & \textrm{if $n=2$}\\ \frac{1}{n(n-2)\alpha(n)}\frac{1}{\lvert x\rvert^{n-2}}=\frac{A}{|x|^{n-2}} & \textrm{if $n\geq3$}. \end{array} \right. $$ Let $f\in C^2_c(\mathbb{R}^n)$. A solution of Poisson's equation is $$ u(x)=\int_{\mathbb{R}^n}\Phi(x-y)f(y)dy. $$ I have tried figuring out a boundedness of $u(x)$ with two cases.
Let $V\subset\subset\mathbb{R}^n$. If $n=2$, fix $\epsilon>0$ then $$ \begin{aligned} |u(x)| & =\frac{1}{2\pi}\bigg\lvert\int_{\mathbb{R}^2}\log\lvert x-y \rvert f(y)dy\bigg\rvert \\ & \leq \frac{1}{2\pi}\left(\bigg\lvert\int_{V\backslash B(x,\epsilon)}\log\lvert x-y \rvert f(y)dy\bigg\rvert+\bigg\lvert\int_{V\cap B(x,\epsilon)}\log|x-y|f(y)dy\bigg\rvert\right) \\ & \leq \frac{1}{2\pi}\bigg\lvert\log\lvert x \rvert \int_{V\cap B(x,\epsilon)}f(y)dy+C\bigg\rvert+M,\quad(\because \textrm{$f$ is bounded and by using DCT.}) \end{aligned} $$ is unbounded as $|x|\rightarrow\infty$.
If $n\geq3$, fix $\epsilon>0$ then $$ \begin{aligned} u(x) & =\int_{\mathbb{R}^n}\frac{A}{|x-y|^{n-2}}f(y)dy \\ & =\int_{B(x,\epsilon)}\frac{A}{|x-y|^{n-2}}f(y)dy+\int_{\mathbb{R}^n\backslash B(x,\epsilon)}\frac{A}{|x-y|^{n-2}}f(y)dy \\ & \leq \|{f}\|_{L^\infty(\mathbb{R}^{n})}\int_{B(x,\epsilon)}\frac{A}{|x-y|^{n-2}}dy+\int_{\mathbb{R}^n\backslash B(x,\epsilon)}\frac{A}{|x-y|^{n-2}}f(y)dy. \end{aligned} $$ The second term on RHS is bounded as $|x|\rightarrow\infty$ since $f\in C^2_c({\mathbb{R}^n})$ and by using dominated convergence theorem. And the first term is bounded since $$ \begin{aligned} \frac{1}{n(n-2)\alpha(n)}\int_{B(x,\epsilon)}\frac{1}{|x-y|^{n-2}}dy & =\frac{C}{\alpha(n)}\int^{\epsilon}_{0}\int_{\partial B}\frac{1}{|rz|^{n-2}}dS(z)r^{n-1}dr \\ & \leq N{\epsilon}^2 \end{aligned} $$
I used change of variables $x-y=rz$ in Line $1$. Therefore, $u(x)$ is bounded as $|x|\rightarrow\infty$ for $n\geq 3$.
I have two questions. One is that whether the change of variables is neccessary, moreover, essential when I taking limits $|x|\rightarrow\infty$ to $\Phi(x-y)f(y)$ as $\Phi(y)f(x-y)$. The other is that if $n=2$, I have learned that $u(x)$ is locally bounded then it could be written as representation formula to Poisson's equation for some $\delta>0$ such that $|x-y|>\delta$?