We consider the following singularly perturbed Schrödinger equation
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.