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

<p style='text-indent:20px;'>In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1a"> \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} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula> is a parameter, id="M2">\begin{document}$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $\end{document}</tex-math></inline-formula>, id="M3">\begin{document}$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) and id="M4">\begin{document}$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $\end{document}</tex-math></inline-formula>. Under some "Berestycki-Lions type assumptions" on nonlinearity id="M5">\begin{document}$ f which are almost necessary, prove that problem id="M6">\begin{document}$ (\rm P) has nontrivial solution id="M7">\begin{document}$ \bar{u}\in H^{1}(\mathbb{R}^{3}) such id="M8">\begin{document}$ \bar{v} G(\bar{u}) ground state of following problem</p><p id="FE1b"> \begin{document}$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$ id="M9">\begin{document}$ G(t): \int_{0}^{t} g(s) ds We also give minimax characterization for id="M10">\begin{document}$ $\end{document}</tex-math></inline-formula>.</p>