Improper integral of $\frac{1}{|x|^p+|y|^q}$

Question

Question

Show that

$$ \int_{D}\frac{1}{|x|^p+|y|^q}, $$

where $D=\{(x,y)\in\mathbb{R}^2\ |\ 0<|x|+|y|\leq1\}$, exists when $\tfrac{1}{p}+\tfrac{1}{q}<1$.

Attempt

So far, I've been trying to bound the function using something similar to the inequality: $(x+y)^n\leq2^{n-1}(x^n+y^n)$ for $x,y\geq0$ and $n\geq1$. However, I'm not sure if there's an analogue for multiple powers.

Any nudge in the right direction would be appreciated

Answer 1

Just consider the first quadrant. The trouble is near the origin.
$$\iint\frac1{x^p+y^q}dxdy$$

Change coordinates to $(u,v)$ with $x^p=u^2,y^q=v^2$, then change again to polar coordinates.