I am a bit confused about Maldacena's original decoupling argument. There are two different low energy (i.e, $\alpha^\prime \to 0$) descriptions of the stack of D3branes:

$\mathcal{N}=4$ SYM and 10D type IIB SUGRA.

Full type IIB superstring in $AdS_5 \times S^5$ and 10D type IIB SUGRA.
Comparing (1) and (2) (actually cancelling 10D SUGRA!) we obtain the celebrated AdS/CFT correspondence. I have the following questions regarding this argument.

If one takes $\alpha^\prime \to 0$ it is same as taking $G_N \to 0$. Then how do the branes backreact to produce nontrivial background namely $AdS_5 \times S^5$?

One arrives at the AdS/CFT correspondence by taking $\alpha^\prime \to 0$, by the above decoupling argument. Then how can one claim that there should be full string theory in $AdS_5 \times S^5$? I understand that any highenergy excitation will be infinitely redshifted for the observer at infinity. But these are all happening at $\alpha^\prime \to 0$!

Isn't full string theory defined only on asymptotically AdS rather than AdS? (I am not sure about this though.)

Also the radius of the $S^5$ turns out to be same as $AdS_5$ scale, $L$. Now small $L$ means highly fluctuating string i.e., quantum gravity regime and thus notion of this classical backgrounds break down. Then how can one do KaluzaKlein reduction of the $S^5$ ?
This post imported from StackExchange Physics at 20150726 09:29 (UTC), posted by SEuser pinu