So this is a common question, but I've spent a while looking into it and I can't seem to find a solution.
I recently bought a PS3 Slim and due to a lack of any spare HDTVs I was planning on hooking up my monitor. I have read that running an HDMI to DVI cable from the PS3 to the monitor and setting the PS3 to HDMI should work fine, and I have also tried using a regular HDMI cable, but my monitor does not display anything. This is especially unfortunate because my monitor has a jack for my headphones which would eliminate any need to purchase audio adapters.
The monitor is HDCP compliant and I have tried switching between the different monitor inputs so that rules out a few possible causes, and it shouldn't be the PS3 because it works on my TV.
Ok this is going to sound hard but it really isn't and sorry if you cant understand this, im not good at saying instructions. You're going to need to connect the ps3 to your tv and the monitor at the same time. Then on the ps3 go to display settings and select video output settings. Up to this point you should be looking at your tv. Then select hdmi as your output device and choose custom settings. Now go though each resolution one at a time selecting and testing. If your screen goes black after selecting one then it wont work on the monitor (for mine it was 1080i that did not work and it might not on yours too). It should most likely go to the highest 1080p and be alright. Once you have set that as the default it will be good. If you have any trouble doing this I can go into more detail.
While that didn't work, I did something similar which did. I had the TV connected and I swapped my current monitor for another monitor with HDMI support. For some reason, this monitor worked without any trouble and I selected the resolutions and finished the set-up. I could then hook my other monitor to the PS3 and it would work without any problems. And you're right; now that the default settings are in, I can switch between my current monitor and the TV without any problems.