Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $R^3$

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where$a,$$b>0$are constants,$2<p<5$and$V:\R^3\rightarrow\R$. Under certain assumptions on$V, we prove that \eqref{0.1} has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results can be viewed as a partial extension of a recent result of He and Zou in [16] concerning the existence of positive solutions to the nonlinear Kirchhoff problem (\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=f(u), u\in H^1(\R^3), u>0, x\in \R^3$, where$\varepsilon>0$is a parameter,$V(x)$is a positive continuous potential and$f(u)\thicksim |u|^{p-1}u$with$3<p<5$ and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,36], which deal with Schr\"{o}dinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem.