We consider the following singularly perturbed Schrödinger equation
where
916302063&_mathId=si2.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=a9acfab3bd89e367be224e0de1709a5c" title="Click to view the MathML source">N≥3,
916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V is a nonnegative continuous potential and the nonlinear term
916302063&_mathId=si4.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=820b20afa69e05ef7d7bffdfa48bb3b0" 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
916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V has a positive local minimum, then for small
916302063&_mathId=si6.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=62c9239737b4e084fdfe6076479ecdb7" 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
916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V as
916302063&_mathId=si8.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=3915f0fbfb7ed26169b9cc7325d86bb9" title="Click to view the MathML source">ε→0. In particular, the potential
916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">V is allowed to be either
compactly supported or decay faster than
916302063&_mathId=si10.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=6d8bc0aa5724428b90ccd5ed9e272027" title="Click to view the MathML source">∣x∣−2 at infinity. Moreover, a general nonlinearity
916302063&_mathId=si4.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=820b20afa69e05ef7d7bffdfa48bb3b0" title="Click to view the MathML source">f is involved, i.e., the
monotonicity of
916302063&_mathId=si12.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=263be07dc382d8710e221ee5f9c9677d" title="Click to view the MathML source">f(s)/s and the
Ambrosetti–Rabinowitz condition are not required.