Generalization of the following equation to higher dimensions

Question

I have the following equation $$\Delta u+cu=0,\,c>0,$$ in $\mathbb{R}^3$. By considering the function $u(x)=\phi(r),$ where $r=|x|$, i end up with the equation $$\phi''\frac{2}{r}\phi'+c\phi=0,$$ and now with the change $\phi(r)=\frac{\psi(r)}{r},$ i get the equation $$\psi''+c\psi=0,$$ so i end up with the solution $\phi(r)=C_1r\cos (\sqrt{c}r)+C_2r\sin (\sqrt{c}r).$ I was wondering what happens to the general equation for $\mathbb{R}^n$. With the same change $u(x)=\phi(r),$ for $r=\sqrt{x_1^2+\cdots+x_n^2}$ we end up with the equation $$\phi''+\frac{n-1}{r}\phi'+c\phi=0,$$ but now the change $\phi(r)=\frac{\psi(r)}{r},$ doesn't vanish the $\phi'$ term. What's the strategy for the general case? Any hint will be very appreciated.

Answer 1

I think you can get rid of the derivative by setting $\psi(r)= r^{(n-1)/2} \phi(r)$. Then $$ \psi'(r)=\frac{n-1}{2}r^{(n-3)/2}\phi(r) + r^{(n-1)/2} \phi'(r), $$ $$ \psi''(r)=\frac{n-1}{2}\frac{n-3}{2}r^{(n-5)/2}\phi(r) + (n-1)r^{(n-3)/2}\phi'(r) + r^{(n-1)/2} \phi''(r) $$ $$ =\frac{(n-1)(n-3)}{4} r^{(n-5)/2}\phi(r) + (n-1)r^{(n-3)/2}\phi'(r) - r^{(n-1)/2}(\frac{n-1}{r}\phi'(r)+c\phi) $$ $$ =\frac{(n-1)(n-3)}{4} r^{(n-5)/2} \phi(r)+c\psi(r) $$ $$ =(\frac{(n-1)(n-3)}{4r^2} +c)\psi(r). $$ However this equation for $\psi$ has not constant coefficients and is singular in $0$. You can get a fundamental system by Bessel functions of the first and second kind.