What would be my case study for this? How can I narrow it?

I am writing a political science research paper. The question is in case of autocratic leaders, when and why they care about public opinion. I can narrow the question, for example by just focusing on specific autocrats or cases in history. I need suggestions on how to narrow it and hypothesis suggestions. Maybe even some examples. I don’t need a detailed answer but a guidance on how to construct this. I thought about the interests of the elite class determines when autocrats care about public opinion. For this, for example, I can compare monarchs vs. citizen autocrats or even religious leadership since elite-class differs. However, I’m not sure if this would work or be a valid argument. 
Another argument is when public opinion promotes protests and it is popular, autocrats have to care about it. Again not sure if it works since it’s too broad and obvious. What would be my case study for this? How can I narrow it?

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