I, personally, would never list at the bottom range.
Firstly is confidence - I have skills the company needs, I know the value of those skills. If I sell those skills short, then the person hiring may doubt how good they are.
Secondly is that the value of your time is not dependent on where you live. Even if you are happy to accept a lower amount due to lower costs of living.
That said - if you are happy to accept near the bottom, I would start at the middle range, or perhaps just below - wait for the negotiation to start - if they are happy to pay that amount, then you've got a little more. If they aren't, then you can haggle down:
"I would like $X, it's the middle of your salary bracket"
"Well, we feel for a candidate like you, that's too high"
"Okay, considering that I'm remote and my costs of living are lower, I could come down to $Y"
"We think $Y is doable! Welcome aboard!"