Removing duplicates from an unsorted linked list
This review is two-fold: once, in case you have to uphold your restriction and then what changes if you allow extra space.

Review


Comments: they are simply missing. You need at least some javadoc, and for non-trivial methods like this some more never hurts. Oh, and skip the usual excuses about how this code is somehow special and doesn't need comments.
Clarity and Intent: if (current.equals(head)) reads like you are comparing the actual nodes, while duplicity is a matter of equal node values. You may have overwritten the equals method accordingly (I assume so for the sake of correctness), but I would prefer to clearly portray your original intent here by comparing the actual values.
Efficiency and readability: As the answer by zdd points out you could save one pointer, but that's kind of pointless. I prefer your version as its much clearer to a reader what current, prev and next are. However,efficiency is not going to get below O(n^2) with this approach.


Efficiency Improvement

You said that you need a solution "without using extra memory". What you really meant of course is a solution "with only a constant amount of additional memory" (your pointers require memory as well after all). What you did not require, however, is that this operation has to leave the list elements' ordering intact. Therefore, a faster approach in O(n log(n)) would be to first sort the list (in-place to satisfy the memory requirement), then simply walk through the list comparing only neighbors (another O(n)). While this does not make much difference of course for smaller lists, the memory requirement indicates much larger lists, and then the difference between O(n^2) and O(n log(n)) may well be the difference between another problem and a solution.