Got a part number/data sheet? Such things as negative resistances can be built of course, they are the basis of "foldback current limiting" in power supplies and audio amplifiers. They're a bit tricky. But don't you want constant current for supplying this chip? Not sure I understand your problem. A sneaky trick used when you can is to use a capacitor, instead of a resistor, for voltage dropping. You can't drive a half-wave diode that way, the cap just charges up, but I've seen this trick used to get a little current at low voltage from the mains with low power losses (since the cap has only imaginary part in the reactive impedance).
Here's a horrible hack Dan Meyer used to try to keep the output transistors in the "tiger" inside their SOA's. Q2 senses the voltage at the emitter resistor of the main output, and the collector voltage, and if the sum through the resistive divider is greater than a Vbe, it shorts the drive to the power transistor....
- SOA.gif (4.08 KiB) Viewed 6162 times
For the other thing, you use a capacitor to the mains, to a series resistor to ground. You get a voltage divider due to the impedances of each at mains frequency, and that smaller AC signal can now be rectified. The regulation is lousy of course, but this trick is often used to power chips that have a built-in zener across their supply terminals. The resistor wastes some power, about equal to the delivered power best case, but for something small and cheap, it works. There are some variations that substitute the resistor with a zener, then a rectifying diode that conducts when the zener is reverse biased (the way it has a normal zener volts across it). I rarely use this trick, but it's out there in a lot of off-mains switchers. Mostly in stuff where the last penny counts, like microwave ovens, other cheap consumer kit, etc. The reason it's not used more is that the series cap has an impedance that is lower the higher the frequency - and line spikes have a lot of HF content, which makes the current through the cap (and therefore the volts across the load impedance) go very high - and things emit the magic smoke.
Posting as just me, not as the forum owner. Everything I say is "in my opinion" and YMMV -- which should go for everyone without saying.