Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold

In this paper, we study the following nonlinear problem of Kirchhofftype:\begin{equation}\label{(0.1)}\left\{%\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \hbox{$x\in \mathbb{R}^3$}, \\ u>0, & \hbox{$x\in \mathbb{R}^3$}, (0.1) \\ \end{array}%\right.\end{equation}where $a,$ $b>0$ are constants, $V:\mathbb{R}^3\rightarrow\mathbb{R}$ and $f(t)$ is subcritical and superlinear at infinity. Under certain assumptions on non-constant potential $V$, we provethe existence of positive high energy solutions by using alinking argument with a barycenter map restricted on a Nehari-Pohožaev type manifold. Our main result has solved Kirchhoff equation (0.1) with superlinear nonlinearities, which has not been studied, and can be viewed as a partial extension of a recentresult of He and Zou in [9] concerning Kirchhoff equations with 4-superlinear nonlinearities.