Question: For which real numbers $p,q$, does the integral:
$$I=\iint_{|x|+|y|\ge1}\frac{1}{|x|^p+|y|^q}dxdy $$
converge?
What I tried so far:
The integrated function, $f(x,y)$, is non-negative.
Based on symmetry arguements, we can look at the integral on the first quadrant of the plane: that is,
$$I'=\iint_{x+y\ge1}\frac{1}{x^p+y^q}dxdy,$$ And then multiply the result by 4 (if it converges).
The domain of integration can be written as a disjoint union (besides a negligible set):
$$E=\underset{E_1}{\{ 0\le x\le 0.5,y\ge 1-x \ge 0.5 \}} \cup \underset{E_2}{\{ 0\le y\le 0.5,x\ge 1-y \ge 0.5 \}} \cup \underset{E_3}{\{ x,y\ge0.5\} } $$
We need $\iint f$ to converge in all three subsets.
If $q>1$, then $\iint_{E_1}f<\infty$, and if $p>1$ then $\iint_{E_2}f<\infty$.
From this point on I'm stuck- I can't figure out the convergence for $E_3$, and I also haven't considered the cases $p\le0$ or $q\le0$ yet (only for $p,q>0$).
How does one approach the rest of the problem?
HINT
We can show that
$$I'=\iint_{x+y\ge1}\frac{1}{x^p+y^q}dxdy\ge \int_1^\infty\frac{1}{x^p+1}dx $$
which diverges for $p\le 1$ and
$$I'=\iint_{x+y\ge1}\frac{1}{x^p+y^q}dxdy\le \int_1^\infty\frac{1}{x^p}dx+\int_1^\infty\frac{1}{y^q}dy+\iint_{x+y\ge1, \, x\le1, \, y\le1}\frac{1}{x^p+y^q}dxdy+\\+\int_0^{\frac \pi 2}\, d\theta\int_1^\infty \frac{r}{r^p\cos^p\theta+r^q \sin^q \theta}dr$$
and the first two integrals converges for $p,q>1$, the third is a proper integral and for the latter assuming wlog $p<q$
$$\int_0^{\frac \pi 2}\, d\theta\int_1^\infty \frac{r}{r^p\cos^p\theta+r^q \sin^q \theta}dr\le\frac \pi 2 \cdot M(p,q)\int_1^\infty \frac1{r^{p-1}}dr$$
where $M(p,q)$ is an upper bound for the function
$$\frac{1}{\cos^p\theta+r^{q-p} \sin^q \theta}$$