As far as I can find by searching the docs & settings, it's not possible to keep the search results open. It is however, very straightforward to reopen them and the original search & results should be preserved. If you want to, you could file a feature request in YouTrack. It would help if you could explain your workflow; what are you trying to achieve that would require the search results to remain open?. Personally I use search to navigate somewhere and would not need to keep them open, so I am trying to understand why you would.