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 $V$. In particular, the monotonicity of $f(s)/s$ and the Ambrosetti-Rabinowitz condition are not required.
Categories: math.AP