stack sizes
You don't tell us the stack sizes--maybe you aren't thinking about them. When I look at Axs and it is raised to me, I think "how much might i win if I do make the nuts?" With 2 cold callers in front, you probably have good enough implied odds, but you should always be thinking about how to win an opponent's entire stack. So the bigger the stack, the more outrageous the draw you can play.
When the flop comes, there are 3 routes. First, you can fold because your kicker stinks. Second, you can flat call. You plan to let him bluff into you repeatedly. The downside is that if you are outkicked, you are going to lose a huge pot (and maybe second guess yourself all night for losing a huge pot on a crappy hand). You must balance this against how much he is likely to lose stealing without an ace. Lastly, you can make a normal (pot size) reraise. It's 20-25 BB pre flop, he bet 10BB, you now raise to 50BB. If he reraises, you can comfortably fold. Notice that he will fold some hands that dominate yours.
The choices here depend on stack sizes. Also, do you have a backdoor flush draw? It can help the second plan (calling) enormously. He doesn't always bet the turn. And if the money is deep enough, you will occasionally win an extra bet on the river. That extra bet is VERY significant at no limit.
Your mini-raise is not amongst my listed choices. Because it screams, "please reraise so I can get away from my hand." Which he did. This guy knows you. So next time you flop a set, give him the mini-raise.
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