Existence of standing waves of nonlinear Schrِdinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ε 2 Δ u − V ( x ) u + u p = 0 , x ∈ R N , we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for p ∈ ( N N − 2 , N + 2 N − 2 ) or nonnegative V satisfies l i m i n f | x | → ∞ V ( x ) | x | 2 log | x | > 0 for p = N N − 2 .