I did not see if this was a new home or older home. However answer wouild be the same.
If it were me I would go to the following link and perform a demand load calculation to find the minimum service size you are required per NEC rules to have serving your home. If you service size is more than minimum calculated by the link below the follow the instructions after the link.
If you service size is adequate then take off your panel cover and read to ground the 120 volt rated voltage of both line one and line two of the main breaker service connections from the meter at the main breaker. Turn on the a/c unit while reading voltage on both lines separatly. If the voltage drops during strating of a/c but returns then your voltage problem is before your main service panel. Then ask the power company to open your meter base. Again test the two incoming lines if this voltage drops again inside the meter the problem is with the power company lines or transformer. You may or may not get the power company to correct if on their lines.
If you find no voltage drop showing on your voltage tester at teh main breaker when the a/c starts then suspect an undersized branch circuit conductor going to your a/c unit.