Find all radial solutions of $\Delta u(\underline{x})=\frac{1}{(1+\parallel x\parallel^2)}$ on $\mathbb R^2\backslash\{0\}$

Question

So far I've written $\Delta u(\underline x)=\Delta u(x_1,...,x_n)$ and I think that this equals this: $$\frac{\partial^2u}{\partial x_1^2}+...+\frac{\partial^2u}{\partial x_n^2}.$$ I also think that $\parallel x\parallel^2=(x_1^2+...+x_n^2)$. Using this information I now have the following equation: $$\sum_{i=1}^n \frac{\partial^2u}{\partial x_i^2}=\frac{1}{(1+x_1^2+...+x_n^2)}$$ but I'm not sure if what I've done is actually useful or what I should do next. Can anyone help?

Answer 1

A radial solution is of the form $u(x)=f(\|x\|)$. You need a formula for $\Delta u$ in terms of $f$. Since using the multivariable chain rule for second derivative is a bit painful, I prefer to derive the formula as follows:

1. The gradient of $u$ is $f'(\|x\|) x/\|x\|$.
2. The flux of the gradient of $u$ across a sphere of radius $r$ is $\omega_{n-1} r^{n-1}f'(r) $ where $\omega_{n-1}$ is the constant equal to the area of unit sphere.
3. On the other hand, the net flux out of spherical shell $r<\|x\|<r+dr$ is the integral of $\Delta u$ over this shell. The Laplacian $\Delta u$ is also a radial function, due to the invariance of Laplacian under orthogonal coordinate transformations. Writing it as $g(r)$, we get the net flux to be $\omega_{n-1} r^{n-1} g(r)\,dr$ up to negligible error term.
4. From 3 and 4, $$\frac{d}{dr}(\omega_{n-1} r^{n-1}f'(r) ) = \omega_{n-1} r^{n-1} g(r)$$ hence $$g(r) = r^{1-n} \frac{d}{dr}( r^{n-1}f'(r) ) $$

Now you can equate $g(r)$ to $1/(1+r^2)$ and solve the differential equation for $f$.