We extend the Abrams-Strogatz model for competition between two languages [Nature 424, 900 (2003)] to the case of n(>=2) competing states (i.e., languages). Although the Abrams-Strogatz model for n=2 can be interpreted as modeling either majority preference or minority aversion, the two mechanisms are distinct when n>=3. We find that the condition for the coexistence of different states is independent of n under the pure majority preference, whereas it depends on n under the pure minority aversion. We also show that the stable coexistence equilibrium and stable monopoly equilibria can be multistable under the minority aversion and not under the majority preference. Furthermore, we obtain the phase diagram of the model when the effects of the majority preference and minority aversion are mixed, under the condition that different states have the same attractiveness. We show that the multistability is a generic property of the model facilitated by large n.