The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term

In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term:

$$\begin{array}{r}-\mathrm{d}\mathrm{i}\mathrm{v}(g^2(u)\mathrm{\nabla }u)+g(u)g^{\prime }(u)|\mathrm{\nabla }u{|}^2+V(x)u=\left(|x{|}^{-\mu }\ast F(u)\right)f(u),x\in {\mathbb{R}}^N,(\mathrm{P})\end{array}$$

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

$$\begin{array}{r}-\mathrm{\Delta }v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))}=\left(|x{|}^{-\mu }\ast F(G^{-1}(v))\right)f(G^{-1}(v)),x\in {\mathbb{R}}^N,(\overline{\mathrm{P}})\end{array}$$

where $G(t):={\int }_0^{t}g(s)ds$. We also give a minimax characterization for the ground state solution $\overline{v}$.