  We study existence of maximizer for the Trudinger-Moser inequality with
general nonlinearity of the critical growth on $R^2$, as well as on the disk.
We derive a very sharp threshold nonlinearity between the existence and the
non-existence in each case, in asymptotic expansions with respect to growth and
decay of the function. The expansions are explicit, using Ap\'ery's constant.
We also obtain an asymptotic expansion for the exponential radial Sobolev
inequality on $R^2$.
