Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:\begin{document}$\left (1\!+\!b\int_{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2} dx \right) \left[-\mathrm{div} \left(g^{2}(u)\nabla u\right)\!+\!g(u)g'(u)|\nabla u|^{2}\right] \!+\!V(x)u\! = \!f( u),(\rm P)$\end{document}where \begin{document}$ b>0 $\end{document} is a parameter, \begin{document}$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $\end{document} , \begin{document}$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $\end{document} and \begin{document}$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $\end{document} . Under some "Berestycki-Lions type assumptions" on the nonlinearity \begin{document}$ f $\end{document} which are almost necessary, we prove that problem \begin{document}$ (\rm P) $\end{document} has a nontrivial solution \begin{document}$ \bar{u}\in H^{1}(\mathbb{R}^{3}) $\end{document} such that \begin{document}$ \bar{v} = G(\bar{u}) $\end{document} is a ground state solution of the following problem\begin{document}$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} dx \right) \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$\end{document}where \begin{document}$ G(t): = \int_{0}^{t} g(s) ds $\end{document} . We also give a minimax characterization for the ground state solution \begin{document}$ \bar{v} $\end{document} .