Reading SMS using Arduino Uno and SIM800L and print on LCD (16x2 alphanumeric) using serial connection NEED HELP
The PP3 battery you are trying to use is far from powerful enough to run a SIM800L. It's barely powerful enough to run an Arduino by itself for more than a couple of hours if you're lucky.

The SIM800L, as the datasheet tells us, must be powered by a supply which can provide at least 2A.


  The power supply range of SIM800L is from 3.4V to 4.4V. Recommended voltage is 4.0V. The transmitting burst will cause voltage drop and the power supply myst be able to provide sufficient current up to 2A.


The Arduino's on-board 5V regulator can't provide more than 800mA if you're lucky (far less if the incoming voltage is more than around 7V due to heat dissipation). Running from USB may allow peaks of up to 2A on a DCP USB socket without the Arduino's PTC fuse blowing.

A typical PP3 battery has an impedance of 1.7 ohms. That means a dead short circuit across the terminals would theoretically allow 9/1.7 = 5.2A maximum ever to flow. That kind of current would dissipate about 46W in heat inside the battery causing it to explode. Also at that current, given the typical low capacity of about 154mAh, the battery would be flat within about 100 seconds.

The maximum rated current for one of these batteries (at least, the maximum current shown on any of the graphs in the datasheet) is 250mA. 

You need to power your SIM800L in a completely different manner, including two important things:


You need a power source that can provide 2A at the rated input voltage of your module (5V?)
That source needs to either be separate from the Arduino (except the ground, which must connect to the Arduino as well as the module) or if it is 5V may be used to directly power the Arduino through the 5V pin.