There is an upgrade guide for Jersey in the WebLogic 12.1.2 documentation but it doesn't cover upgrading to Jersey 2.x. That is exactly what I wanted to do however, the process is very similar to the 1.x upgrade.
My approach to using Jersey 2.x was to first package it as a shared library. I've previously covered how to package shared libraries for WebLogic deployment so will not cover it again here. How you package Jersey depends on the features you use, but in the minimum the following is required as a dependency (change version as required) - Servlet based server-side application...
You're likely going to want other parts of Jersey as well, the modules list contains these. I had to use a number of media modules so I added those to my dependency list, e.g.
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