Is there any active GSM antenna exist?
&quot;Active antennas&quot; include an amplifier to improve reception.
This works well with things that only receive.  GPS receivers and television receivers only receive.  It is fairly easy to include an amplifier to make reception better.
GSM has to transmit as well as receive.  It is much more difficult to make an active antenna if you have to be able to transmit through it.
If the communication schemes were simplex (either transmit or receive but never both at the same time) then you could include an automatic switch in the active antenna to disconnect the receiver amplifier when transmitting.
Phones aren't simple, nor do they use simplex communication.  They transmit and receive at the same time.
Anything that can seperate the receive and transmit signals is already half of a phone.
Given that, there's not much point in having an active antenna.  Your phone still has to split things to transmit and receive, so you might as well put any gain there (on the phone PCB) instead of the antenna.  That keeps the antenna simple and cheap.

As to solving the actual problem:
There might not be much you can do.
You could use an antenna with gain.  These are directional antennas that receive and transmit in one direction better than in other directions - the worse it receives to the back side, the better it receives to the front side.
The problem there is that consumer devices are limited in the amount of power they are allowed to transmit.  A directional antenna can take you over the limit in its main direction.  The power the phone transmitter sends stays the same, but it is focused in one direction.  Kind of like you can look at brightly lit wall, but a focusing mirror of the same size in the same light would be blinding.
See if an omnidirectional antenna with gain helps.  These get their gain by being taller than normal.  They have poorer reception upwards, but better reception from all directions around them.  Kind of like a donut shape.
Something like this:

Mount it in the center of a large, flat piece of sheet steel.  Say, a square or circle about 1 meter across.  Mount the whole thing outside the building, and on top (up high.)  You'll need to see about proper grounding and lightning protection - if you don't, you can expect a lighting strike to take your system out of operation if the safety inspectors don't.
(BTW: That's probably not the right antenna.  It is just an example.  It probably has the wrong connector, and probably isn't in the correct band.)