When to give up and ask the research community? While reflecting on my last research paper (currently in the hands of a referee) I discovered a closely related pair of problems that, as far as I can tell from published record, nobody else has considered.  


  For those interested in topology, the problems concern a property of
  first-category continuous images of the half-open interval of reals. 
  The property has previously only been of interest in compact spaces,
  but I want investigate for other classes of spaces.


A solution to one or both would make for a nice paper I think.  I've been thinking about them for a couple of weeks with basically no progress.  

Should I just tack the problems at the end of my research paper (give up?), or keep them to myself and hope to eventually solve one?
This is a type of question that you could ask on MathOverflow.  

In my experience, one of the best uses of MO is if you have a "research level" question that (i) you want answered, and you're not particular about how and (ii) you think "Probably[/maybe/I wonder if] someone with the right expertise could solve this almost immediately."  

If both of the above apply, I would encourage you to ask on MO.  On the other hand, if you want to keep maximal control over your involvement with the question and its solution, less wide disbursement is probably best.  (Most of my own MO questions have been things that come up in my research but are not exactly the questions I am trying to solve in order to write my next paper.) If you really see yourself getting a paper out of these questions, I would vote for spending more than two weeks thinking them over.