Asymptotically linear Schrödinger equation with potential vanishing at infinity

We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation where the potential V(x) may vanish at infinity, f(s) is asymptotically linear at infinity, that is, f(s) O(s) as s→+∞. For this kind of potential, it seems difficult to find solutions in , i.e. bound states of (0.1). If f(s)=sp and p (σ,(N+2)/(N−2)) with σ 1, Ambrosetti, Felli and Malchiodi [A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005) 117–144] showed that (0.1) has a solution in and (0.1) has no ground states if p is out of the above range. In this paper, we are interested in what happens if f(s) is asymptotically linear. Under appropriate assumptions on K, we prove that (0.1) has a bound state and a ground state.