Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity

We study the following nonlinear Schrödinger equation −Δu+V(x)u=K(x)f(u),  x∈ℝN,  u∈H1(ℝN), where the potential V(x) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem () under a Nahari type condition. Furthermore, if V(x),K(x) are radically symmetric with respect to x∈ℝN, it is shown that problem () has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if f(u)=up, then the growth restriction σ≤p≤N+2/N-2 in Ambrosetti et al. (2005) can be relaxed to σ~≤p≤N+2/N-2, where σ~<σ if 0<β<α<2