Have the pcg-k35 and only works when power supply

Check your mains adapter plug that connects into the laptop power socket with a multi-meter, the voltage should be slightly higher than the voltage that is printed on the label on the bottom of the adapter. If the voltage is zero or way below the voltage that is printed on the label then the power adapter is faulty and needs to be replaced.

If the adapter is OK then connect it to the laptop and power it up, then look at the power/battery charging LED light and wriggle the power plug in the laptop's power socket. If the power/battery LED light flashes and in a certain position this LED light stays firm, then the power socket is either faulty or it has a dry solder joint where it is connected to the motherboard.
This can be repaired but it requires the laptop to be completely dismantled to get at this power socket and repair. I suggest you get a quote first.

First, your battery is discharged and most likely bad. The problem with the power cord and adapter could be a loose power connector on the computer or short in the wire itself. If you have access to another adapter try the other adapter with the battery removed and see if you have that same problem. If you do then you know the problem is on the motherboard connector. If not you know it is the Power adapter.