In this paper, we revisit a well-known differential game model to investigate the R&D activity for Greentech innovation in a duopoly market with network externality. In our study, firms carry out independent R&D or Cartel R&D for Greentech innovation which aims to reduce end-of-pipe emission and network externality effects play a role in the R&D activity via the inverse demand functions. Firms are concerned not only with their own profits, but also with the relative profits. At the steady-state equilibrium, the efforts for Greentech innovation under independent R&D are higher than that under Cartel R&D when the degree of altruism is higher than the degree of technological R&D spillover, and the network size under independent R&D is lower than that under Cartel R&D. Moreover, under a given market structure, whether profits under independent R&D are higher than under Cartel R&D depends on the initial network size, which means that at the initial moment firms can decide to undertake independent or Cartel R&D activities based on the initial network size.