The fact that there is no solution is a perfectly valid answer to a question. If that's the conclusion you have reached after extensive research, you should post that as an answer and then accept it (when the system allows you to do so). Or don't: you have no obligation to accept an answer. But either way, there is nothing wrong with answering your own question; in fact, it is encouraged.
These types of answers become even better if they explain (1) why the problem is unsolvable, and (2) what alternative or workaround you can (or have chosen to) use instead.
I know you already said you don't want to delete the question, but I'd just like to point out that just because something is unsolvable right now doesn't meant that it always will be. If/when things change, a new answer could be posted with the solution and you could change which one you've accepted. Besides that, you could be wrong about it being an impossible problem. Someone could always come along who knows something you don't. I've been surprised many times in the past.