Some Existence Results on a Class of Generalized Quasilinear Schrödinger Equations with Choquard Type

In this paper, we study the generalized quasilinear Schr\ {o}dinger equation \begin{equation*} -\text{div}(g^2(u)\nabla u)+g(u)g (u)|\nabla u|^2+V(x)u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u,\ \ \ x\in\R^N, \end{equation*} where $N\geq3$, $0 lt;\alpha lt;N$, $\frac{2(N+\alpha)}{N} lt; p lt;\frac{2(N+\alpha)}{N-2}$, $V: \R^N\rightarrow\R$ is a potential function and $I_{\alpha}$ is a Riesz potential. Under appropriate assumptions on $g$ and $V(x)$, we establish the existence of positive solutions and ground state solutions.