What are possible solutions when promoting competition is not applicable?
There is one universal "solution" to monopolism without government intervention.

Just allow the monopoly to do its worst. It will give very strong incentives to find ways to satisfy needs of potential clients of the monopoly in ways that would diminish market force of the monopoly. Basically it's bet that human creativity will eventually tame (or even kill) the monopoly. The most radical example of such creativity is a creation of an invention that makes the whole monopoly obsolete. Another example of creativity: It's also possible that different people buy services/goods of the monopoly for different reasons, so there would be strong incentives to develop imperfect substitutes for different categories of potential clients of the monopoly. Of course there are many more other ways to outsmart the monopoly, I just gave some abstract examples of it.

Real life example: US shale oil boom as response to monopoly of OPEC