In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{ equation*} -(\varepsilon^2a+\varepsilon b\int_{ \mathbb{ R}^3}|\nabla u|^2dx)\Delta u+V(x)u=P(x)|u|^{ p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{ R}^3, \end{ equation*} where $\varepsilon>0$ is a small parameter, $a, b > 0$ are constants, $p\in(4,6)$ and $V, P, Q$ are potential functions satisfying some competing conditions. We prove the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials $V,P$ and $Q$ as the concentration position of these ground state solutions as $\varepsilon\to0$.
Categories: math.AP