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:

$ \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) $

where $ b>0 $ is a parameter, $ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $. Under some "Berestycki-Lions type assumptions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{3}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem

$ -\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}) $

where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.