My Note 4 battery took a dump after Lollipop. I had to factory reset it to get it back to normal. Turned out to be a few apps that were fine on KK but were not supported under lollipop. Those apps do not appear in the app store anymore.
I've been trying to google my issue with Lollipop but to no avail:
I use a Sony Z3, and after the OTA push to Lollipop, multitasking has taken a kind of nosedive. I used to be able to stream youtube (radioyoutube app) videos for the audio, in the background, while I'm sending out e-mails, surfing the web, whatever. Now, as soon as I do this, the minimized app (radioyoutube) starts stuttering, and the audio becomes terrible, and the app crashes.
I've googled 'stuttering on lollipop' and there are some threads about needing to clear out the cache, but so far the update doesn't seem so positive to me. Maybe it's what you say - that the apps I'm using aren't supported under lollipop, but if so that's a shame.