Atmega328p shutdown under 3.4v

Hello,
I have a problem with a atmega328p used standalone with 3.3v and the 8MHz internal oscillator.
I´m using the board setting “ATmega328 on a breadboard (8MHz internal clock)“ as available on the arduino homepage breadboard-1-6-x.zip.
Powered with 5v the atmega works fine but with voltages under 3.4v it shuts off. Assuming it is the fault of the brown out detection (which would be strange because of the detection steps at 1.8,2.7,4.3 volts) I disabled it by setting the extended fuses to 0x07. This has no effect. Maybe you can help me finding a solution to this problem

How's it wired?

Are you powering it through a regulator? Regulators have a "drop out" - you lose at least a volt across most regulators, so if you're putting 3.4v into a 3.3v regulator, you're probably only getting 2.x volts out, probably bringing the chip out of spec.

Do you have the decoupling caps in place (2 0.1uf caps, one between Vcc and Gnd, other between AVcc and Gnd, ceramic caps, located right next to the chip)? Some guides omit them, despite the fact that the '328p doesn't function reliably without them - and lowering the operating voltage would magnify the effect of this.

Thanks for the fast reply
For testing Im powering the atmega with a lab bench power supply. Also I checked the voltage with a multimeter. Indeed the decoupling caps were not in place. I connected them but they had no effect. (Is the atmega completely off if the problem is caused by the missing caps?) Under 3.4v the atmega has no output voltage on any digital or analog pin.