Look for a small, white IC (ant size) with a "P" in the middle of it. You may need a magnfying lens to see the "P". It should be near the LCD connector. Check it with a multi-meter. Set it to Ohm mode, and put one side of the probe on each side. If the "fuse" (technically called a filter) is good, it should read all 0's. Make sure the battery is disconnected and power is unplugged before doing anything.
If the one there is anything like the one here (mall kisok), I would not have them try a glass only repair. This is a repair that takes skill and equipment to do right. Look for a well rated, established, locally own repair shop. Make sure they cover the screen in case it is damaged during the repair.
From a purely technical stand point, yes it is possible... But it would require you to remove the glass and replace the flex cables (requires a very expensive machine). From a practical standpoint, no.
Have you tried a new home button extension flex cable (the cable that runs from the top of the metal plate to the bottom)... I work on these every day (own a repair shop), and at least once a week I damage one of these cables without knowing it at first. They tend to be the most fragile on the bottom end, and if not taken off carefully the small metal plate under the bottom connector comes off.