In this work, we study the following Kirchhoff equation $$\begin{ cases}-\left(\varepsilon^2 a+\varepsilon b\int_{ \mathbb R^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{ q-1},\quad u>0,\quad x\in { \mathbb{ R}^{ 3}},\\u\to 0,\quad \text{ as}\ |x|\to +\infty,\end{ cases}$$ where $a,b>0$ are constants, $2<q<6$, and $\varepsilon>0$ is a parameter. Under some suitable assumptions on the function $Q(x)$, we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of $Q(x)$ for $\varepsilon>0$ sufficiently small, by using the Lyapunov-Schmidt reduction method. We extend the result in (Discrete Contin. Dynam. Systems 6(2000), 39--50) to the nonlinear Kirchhoff equation.
Categories: math.AP