Find for which $\alpha >0$ this function is in $L^2(\mathbb{R^2})$

Question

For which values of a real parameter $\alpha >0$, the function $$\frac{1}{|x|^{\alpha} + |y|^{\alpha} +1}$$ is in $L^2(\mathbb{R^2})$?

I am pretty sure this involves using polar coordinates (for example it works for the case $\alpha =2$), but all my attempts turned out to be very messy. Any help?

Answer 1

Hint: due to symmetry, it's enough to consider the integral only in the first quarter. Moreover, it is enough to consider the area $x^\alpha+y^\alpha\geq1$. Make a replacement $x=r^\frac{2}{\alpha}\cos^\frac{2}{\alpha}\varphi$, $y=r^\frac{2}{\alpha}\sin^\frac{2}{\alpha}\varphi$