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