Try using the Reduce function and set the partial derivatives both to zero and the parameter a>0.

Like this:

Reduce[{D[fxy, x] == 0 && D[fxy, y] == 0 && a > 0}, {x, y}, Reals]

I think the maximum is going to be when y = 1/(a*x) and x obviously cannot be zero. An example would be 1/e when a=1.

See if this works for you.