I would really appreciate if you could help me to solve this integral or at least determine the bound since I do not know if it converge or not of the integral:
$\iint\limits_R {Exp\left[ { - \max \left( {\frac{a}{x},cy + \frac{d}{x}} \right)} \right]dxdy}$ with $R$ is the region $x \geqslant 0,y \geqslant 0$
and $a,c,d > 0$
For ease of effort I also found out that $\max \left( {\frac{a}{x},cy + \frac{d}{x}} \right) = \frac{a}{x}$ if $c > 0,x > 0,y \leqslant \frac{{a - d}}{{cx}}$
and $\max \left( {\frac{a}{x},cy + \frac{d}{x}} \right) = cy + \frac{d}{x}$ if $c > 0,x > 0,y \geqslant \frac{{a - d}}{{cx}}$
Both $\frac ax$ and $cy+\frac dx$ yield divergent improper integrals, so $\max\{\frac ax, cy + \frac dx\}$ also yields a divergent integral.
[edit: the integrand has now an exponential]
Even with the exponential, the integral is still divergent. In general, $$ e^{-\max\{\frac ax, cy +\frac dx\}} = \min\{e^{-\frac ax},e^{-cy-\frac dx}\}. $$
we know that our integral ($I$) satisfies $$ I \ge \int_0^{+\infty} \int_0^{+\infty} e^{-\frac ax} dx dy, \quad I \ge \int_0^{+\infty} e^{-cy} \int_0^{+\infty} e^{-\frac dx} dx dy $$
Since both these integrals are divergent, so is the initial one.