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 equationequation(0.1){−Δu(x)+V(x)u(x)=K(x)f(u),x∈RN,u∈H1(RN),u(x)>0,N⩾3, where the potential V(x)V(x) may vanish at infinity, f(s)f(s) is asymptotically linear at infinity, that is, f(s)∼O(s)f(s)∼O(s) as s→+∞s→+∞. For this kind of potential, it seems difficult to find solutions in H1(RN)H1(RN), i.e. bound states of (0.1). If f(s)=spf(s)=sp and p∈(σ,(N+2)/(N−2))p∈(σ,(N+2)/(N−2)) with σ⩾1σ⩾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 H1(RN)H1(RN) 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)f(s) is asymptotically linear. Under appropriate assumptions on K, we prove that (0.1) has a bound state and a ground state.