In markets where consumers have switching costs and firms cannot price‐discriminate, firms have two conflicting strategies. A firm can either offer a low price to attract new consumers and build future market share or a firm can offer a high price to exploit the partial lock‐in of their existing consumers. This paper develops a theory of competition when overlapping generations of consumers have switching costs and firms produce differentiated products. Competition takes place over an infinite horizon with any number of firms. This paper shows that the relationship between the level of switching costs, firms' discount rate, and the number of firms determines whether firms offer low or high prices. Similar to previous duopoly studies, switching costs are likely to facilitate lower (higher) equilibrium prices when switching costs are small (large) or when a firm's discount rate is large (small). Unlike previous studies, this paper demonstrates that the number of firms also determines whether switching costs are pro‐ or anticompetitive, and with a sufficiently large (small) number of firms switching costs are pro‐ (anti‐) competitive.