Summary: We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation
where the potential may vanish at infinity, is asymptotically linear at infinity, that is, as . For this kind of potential, it seems difficult to find solutions in , i.e. bound states of (1). If and with , A. Ambrosetti, V. Felli and A. Malchiodi [J. Eur. Math. Soc. (JEMS) 7, No. 1, 117–144 (2005; Zbl 1064.35175)] showed that (1) has a solution and (1) has no ground states if is out of the above range. We are interested in what happens if is asymptotically linear. Under appropriate assumptions on , we prove that (1) has a bound state and a ground state.