Consider the improper double integral$$\iint_\Omega \frac 1{x^p+y^q}\ dx \, dy,$$ here $p,q\in \mathbb{R},$ and $\Omega=\left\{(x,y)\mid 0<x\leqslant 1, 0\leqslant y\leqslant 1-x\right\}.\ $
I want to determine the range of $\ p,q$ , such that the integral above is convergent. But I can only derive that when $p>2$ and $q>2$, this integral is diverge. I think it requires some analytical skills to solve this question, which (unluckily) I don't have. Any help would be appreciated.
A start: Notice right away that if either of $p,q \le 0,$ then the integrand is bounded above by $1,$ so the integral converges.
For $p,q >0,$ the integrand is bounded on the triangle $\{0<x<1, y\ge 1-x\}.$ Thus our integral converges iff
$$\tag 1 \int_0^1\int_0^1 \frac{1}{x^p+y^q} \,dy \, dx <\infty.$$
Let $y=x^{p/q}t.$ Then $(1)$ equals
$$\tag 2 \int_0^1 x^{p/q-p}\int_0^{x^{-p/q}}\frac{dt}{1+t^q}\, dt\, dx.$$
The inner integral in $(2)$ is bounded below by $\int_0^{1}\frac{dt}{1+t^q}\, dt.$ Thus a necessary condition for convergence is that $p/q-p>-1.$ Is it sufficient?