Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2\alpha-2}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2\gamma-2}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $\rho_1,\rho_2> 0$ and $\rho_1+\rho_2=1$, we have $i^\alpha+j^\beta\ge c i^{-\alpha\rho_1}j^{-\beta\rho_2}$ (now $g(x,y)= x^{-\rho_1\alpha}y^{-\rho_2\beta}$ is homogeneous of degree $-(\rho_1\alpha+\rho_2\beta)$, and an integral approximation argument applies provided $\alpha\rho_1\in (1/2,3/4)$, $\beta\rho_2\in (1/2,3/4)$), we should have $\gamma<\rho_1\alpha+\rho_2\beta$.