We consider the following singularly perturbed Schrödinger equation
where
N≥3,
94a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V is a nonnegative continuous potential and the nonlinear term
b20afa69e05ef7d7bffdfa48bb3b0" title="Click to view the MathML source">f is of critical growth. In this paper, with the help of a truncation approach, we prove that if
94a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V has a positive local minimum, then for small
b4e084fdfe6076479ecdb7" title="Click to view the MathML source">ε the problem admits positive solutions which concentrate at an isolated component of positive local minimum points of
94a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V as
ε→0. In particular, the potential
94a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V is allowed to be either
compactly supported or decay faster than
∣x∣−2 at infinity. Moreover, a general nonlinearity
b20afa69e05ef7d7bffdfa48bb3b0" title="Click to view the MathML source">f is involved, i.e., the
monotonicity of
f(s)/s and the
Ambrosetti–Rabinowitz condition are not required.