This depends on many things. Most important is the significance of the questions themselves and whether the partial solutions add anything beyond standard techniques. The questions can be "challenging" without being significant, of course. Do they lead to other things, for example.
If your solution techniques are standard, then they won't add much either, but if you have developed a new way of looking at such problems, then it could be very important.
"Almost half a year" doesn't sound very long, actually, for the stage you are at.
Your advisor should give you good advice here. He or she is more familiar with your problem and your attack and degree of attainment.
However, I come down a bit on the side of waiting to publish. For one thing, if you publish partial results it may lead others to join the search for solutions in competition with you, perhaps denying you a more significant publication (or even a dissertation) in the future.
If it makes sense to separate them, having four or five papers, each with a complete solution would be much better for you than rushing now. The publishing venue is likely to be more prestigious, and also more likely to accept complete results.