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A complete solution to the quaternionic contact Yamabe equation on the qcsphere of dimension $4n+3$ as well as on the quaternionic Heisenberg group isgiven. A uniqueness theorem for the qc Yamabe problem in a compact locally3-Sasakian manifold is shown.

Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group