Intermediate Case of Bertrand and Cournot
These papers could be interesting to you. First, a classical contribution:
Singh Nirvikar and Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, vol. 15(4), pages 546-554.
And these two interesting papers, using the concept of competition toughness to reconcile Cournot and Bertrand:
d'Aspremont, Claude &amp; Dos Santos Ferreira, Rodolphe, 2009. "Price-quantity competition with varying toughness," Games and Economic Behavior, 65, 62-82.
d'Aspremont Claude and Rodolphe Dos Santos Ferreira &amp; Louis-André Gérard-Varet, 2007. "Competition For Market Share Or For Market Size: Oligopolistic Equilibria With Varying Competitive Toughness," International Economic Review, 48, 761-784.