woke up this morning and mine is doing the exact same thing your's is. i had it charging last night, woke up, saw it was on android splash unplugged it and now the phone doesnt stay powered on in either recovery mode and when i try to go to regular mode, it starts to load the home screen then shuts off again. if i dont have the charger plugged in, it doesnt even power on. no lights. no nothing.
my phone is rooted with custom rom but i dont think this has anything to do with it because the phone doesnt power on unless i have the charger attached. i dread calling VM for anything. i did manage to get the phone to stay on for a while in home screen but it only lasted about 10min before shutting off again and doing the whole boot loop dance again. i did notice however that altho i had the phone charging last night, the battery still shows dead as if i left it off the charger all night when i hadn't.
btw u can get it in recovery mode by holding down home+vol down+power first, and then plug in the charger cable. if u plug it in before then it will jus carry u thru the whole loop all over again. i've googled this problem and i've come across a few people who have this same problem when rooted with custom rom or stock unrooted rom. i honestly think its a defect in the craftmanship of the phone regarding the power.
currently the phone stuck in that loop again so i've put it in recovery mode and i'm just going to let it sit there and try and charge. while the phone was cold this morning it now feels warm so i think its actually trying to charge now.
i think the moral to the story with these LG Optimus is you can never let the battery fully die or discharge ever. they have a hard time coming back up. As soon as you get the charge battery warning, it's best to do it or turn the phone off completely until you can or this is what happens.