Is the capacity achieving input of a discrete memoryless channel unique?
I believe the answer is "yes", when $Q$ is square and nonsingular, and "not necessarily", otherwise. See "On the Capacity of a Discrete Channel. I", by Saburo Muroga, Journal of the Physical Society of Japan, Vol. 8, No. 4, July - August 1953, pp 484-494.

You might also be interested in my note on Muroga's paper, in which I attempt to provide an alternate proof of the feasibility of the optimal input probability distribution, in the special case of a binary channel, and to correct what I think is a trivial error in Muroga's paper (although this might simply have been Muroga being too polite to point out a trivial mistake by Shannon), and I also try to simplify a part of Muroga's argument:

"On the Computation of the Shannon Capacity of a Discrete Channel with Noise", S. Cowell, arXiv:1701.08731v2 [cs.IT] [here]1