Existence of competitive equilibrium between max utility function and min utility function
Indeed, there is no competitive eq. To see this case-by-case analysis is the best way forward. 

If either price is 0, consumer 1 will demand an infinite amount, markets will not clear. If prices are both positive, consumer 1 will only ever demand 1 of the 2 goods. But, consumer 2 demands an equal amount of each good is such any such case. No allocation is compatible with both these demands.

The failure lies in the fact that the demand functions (in particular of consumer 1) are not continuous. This happens because 1 has a convex utility function.