Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth

We consider the following singularly perturbed Schrödinger equation −ε2Δu+V(x)u=f(u),u∈H1(RN), where N≥3, V is a nonnegative continuous potential and the nonlinear term f is of critical growth. In this paper, with the help of a truncation approach, we prove that if V has a positive local minimum, then for small ε the problem admits positive solutions which concentrate at an isolated component of positive local minimum points of V as ε→0. In particular, the potential V is allowed to be either compactly supported or decay faster than ∣x∣−2 at infinity. Moreover, a general nonlinearity f is involved, i.e., the monotonicity of f(s)/s and the Ambrosetti-Rabinowitz condition are not required.