I'm not sure where this will go, but lets start by expanding the integrals, as $\displaystyle X$ and $\displaystyle Y$ are independent:
$\displaystyle E_Y\left[ E_X \left[ \chi_{\left\{ X>\alpha Y\right\}} X \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha y}^{\infty} x \ p_X(x)\ p_Y(y)\ dx dy$
and:
$\displaystyle E_X\left[ E_Y \left[ \chi_{\left\{ Y>\alpha^{-1} X\right\}} Y \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha^{-1} x}^{\infty} y \ p_X(x)\ p_Y(y)\ dy dx$
Now we can observe that the region over which we are integrating in the first integral is the part of the x-y plane below $\displaystyle y=\alpha^{-1} x$ and that in the second integral is that part of the plane above the same line.
Quite where to go from here I don't know
RonL