Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

In this paper, we study the existence of positive solutions for the following quasilinear Schrodinger equations \begin{document}$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $\end{document} where \begin{document}$ \kappa>0 $\end{document} , \begin{document}$ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document} is superlinear at infinity, the potentials \begin{document}$ V(x) $\end{document} and \begin{document}$ K(x) $\end{document} are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful \begin{document}$ L^{\infty} $\end{document} -estimates. For the subcritical case ( \begin{document}$ \mu = 0 $\end{document} ) we can deal with large \begin{document}$ \kappa>0 $\end{document} . For the critical case we treat that \begin{document}$ \kappa>0 $\end{document} is small.