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Costa, Jianjun Zhang, Jo\\~ao Marcos do \\'O","submitted_at":"2017-03-15T12:24:42Z","abstract_excerpt":"We are concerned with the following Kirchhoff type equation $$-\\varepsilon^2 M \\left(\\varepsilon^{2-N} \\int_{\\mathbb{R}^N} | \\nabla u|^2\\, \\mathrm{d} x\\right) \\Delta u+V(x)u = f(u),\\ x \\in \\mathbb{R}^N,\\ \\ N\\ge2, $$ where $M \\in C(\\mathbb{R}^+,\\mathbb{R}^+)$, $V\\in C(\\mathbb{R}^N,\\mathbb{R}^+)$ and $f(s)$ is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of $V$ as $\\varepsilon\\to 0$ under certain conditions on $f(s)$, $M$ and $V$. 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