Summary: We consider a class of homogeneous Cournot oligopolies with \((-1)\)-concave price function. We show some useful properties of the revenue function in case of \((-1)\)-concave price function and prove the existence of an equilibrium in the continuous and non-differentiable case. A simple proof of an equilibrium uniqueness result in the smooth case with \((-1)/N\)-concave (\(N\)-number of the firms in the market) price function is provided.