Notebook problem after Vista installation

I installed Windows Vista Home Prmium Express Upgrade on my HP Pavillion
DV6000. I followed all the steps recommended by HP+MS. After this
installation I am having the following problem:
1) If the computer is not plugged to a power outlet (using the internal
battery) it suddenly shuts down. It happens "out of the blue". Some times I
am using Internet, sometimes I am writing a MS Word text, etc... It is
completely unexpected. I have to turn if on again mannually and it starts
normal, without any error message,
2) If I have my computer connected to the power outlet it won't power off,
but it will blink. Nothing else happens, no error message, no problems with
the application, and I just keep on using it as if nothing happened. But I
think this "blinking" is related with the shutting down problem I have when I
am using the battery, i.e., if I was using the battery, the computer would've
shut down itself and not only blinked.
Any clues what should I do to correct this problem?