LCD Power Problems
A lot of LCD problems (turning on for a few seconds and then off again) typically relate to issues with the capacitors on the power board. If you are having problems with your TV turning off randomly, or not turning on at all then the best place to start looking is at the power board.
Take the case off the back of the TV/LCD , and find the PSU/Power board, it will be the board where the power cable from the wall connects to, be careful at this point, as the capacitors on the board might still have residual charge in them.
Have a look at the board and check all of the capacitors, they should have a completely flat top and should not be leaking any liquid. If any of them are bulging or have liquid leaking from them then these will need to be replaced in order to get your set to function again.
If you want to be sure of which capacitors are faulty - you will need to use an ESR Meter to check which capacitors have a high equivalent series resistance, as some capacitors can have faults without displaying any of the above symptoms.
If you are comfortable with electronics and soldering, you can swap the capacitors out - taking care to make sure that the capacitance of the replacements is the same (you can use slightly higher voltage capacitors, if they fit). Given the low cost of capacitors it's usually worthwhile to swap out all of the capacitors on the power board for good measure.
How To Swap Capacitors