In Evan's PDE . We find a solution $u$ of the Laplace's equation in ${\bf R}^{n}$ having this form $u(x)=v(r)$ , where $r=|x|=(x^{2}_{1}+x^{2}_{2}+\cdots+x_{n}^{2})^{1/2}, n\ge 2~.$
Now , the following is my working to write in detail for solving the so-called Laplace's equation and some question .
Question : Am I right to separate two cases to discuss the condition of absolute value $?$ and Is my working valid $?$
Working :
By simple computation ,precisely the chain rule , we obtain $$v''(r)+\frac{n-1}{r}v'(r)=\Delta u=0$$
If $v'(r)\ne 0$ ( otherwise $v(r)$ is a constant ) then we have $$\bigg(\ln|v'(r)|\bigg)'=\frac{v''(r)}{v'(r)}=\frac{1-n}{r}$$
Then by FTC (fundamental Calculus Theorem) , we have for some $\delta >0$ that $$\ln|v'(r)|-\ln|v'(\delta)|=\int_{\delta}^{r}\bigg(\ln|v'(\alpha)|\bigg)'d\alpha=\int_{\delta}^{r}\frac{1-n}{\alpha}d\alpha$$
Hence $$\ln|v'(r)|-\ln|v'(\delta)|=(1-n)\big(\ln r-\ln \delta\big)=\ln r^{-(n-1)}-(1-n)\ln\delta$$
then $$\ln|v'(r)|=\ln r^{-(n-1)}+\ln\delta^{n-1}+\ln |v'(\delta)|=\ln r^{-(n-1)}+\ln\delta^{n-1}|v'(\delta)|$$
Then $$|v'(r)|=e^{\ln r^{-(n-1)}}\cdot e^{\ln |v'(\delta)|\delta^{n-1}}=r^{-(n-1)}\cdot \ln |v'(\delta)|\delta^{n-1}=\frac{C}{r^{n-1}}$$
, where $C=\ln |v'(\delta)|\delta^{n-1}$ .
Now if ${v'(r)>0}$ then we have $$v'(r)=|v'(r)|=\frac{C}{r^{n-1}}$$
while if ${v'(r)<0}$ , thus one has $$-v'(r)=|v'(r)|=\frac{C}{r^{n-1}}\Longrightarrow v'(r)=\frac{-C}{r^{n-1}}$$
Thus we conclude that $v'(r)=\displaystyle\frac{K}{r^{n-1}}$ for some constant $K$ .
Whence, if $r>0$ then we obtain the solution $v(r)=b\ln r+ c$ for $n=2$ and $v(r)=\frac{b}{r^{n-2}}+c$ for $n\ge 3$ , where $b,c$ are constants .
Any comment or valuable suggestion I will be grateful . Thanks for considering my request .