I updated the question.

It is a linear problem, but that tells us nothing about the structure of the optimal transportation plan and other properties of the optimal cost. For example, for a 2D problem, what is the transport plan looks like?

Thank you. Your reference is very helpful. In Section 10.4 of Viana's book, it seems that $d$ has to be positive to satisfy the condition of the theorem. The non-trivialness of the problem you mentioned is because of the negative power of the distance, which causes the cost $c(x,y)$ to decrease when $\vert x-y\vert $ increases. But it is still positive, therefore the infimum is always non-negative. However, the problem in my setting is negative cost.The proof from $d$ is always positive to $d$ is always negative is nontrivial. For example, the infimum may not exist if $d$ is negative.

@YuvalPeres Yes, I have changed the notation. Actually if we only consider the measure that has a density function, then the optimal transport or maximum cost transport could be seen as it is defined on functional space. That is the reason why I wrote $M_c(f,f)$ in the first place. Sorry for the inconvenience.