Computer powers on, but does not boot

Hello,
I've had my desktop computer for about three years. It is a LG. One it show "core dumped" & automatically shutdown & do not start again.
The power button usually lights up, but now it does not. The DVD drive still opens, and a few lights blink on start up as usual.
The screen is blank. The power supply fan runs, the other fan runs too.
I have made sure everything is plugged in, and tried plugging it into different outlets. I can't boot from a CD, the keyboard turn on and the mouse blinks for once.
As far as starting up goes, everything sounds normal. No beeps or clicks to hint at hard drive failure.
Sometimes when I try and try to restart it, it just starts and run well until I shutdown it again. If I put it into sleep mode it not start again. But if I restart it again and again there is no problem.
If everything seems to be going, what is stopping it from booting up?