Ground states for a class of quasilinear Schrödinger equations with vanishing potentials

In this paper, we study a class of quasilinear Schrödinger equation of the form \begin{document}$ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2}))u = K(x)|u|^{q-2}u,\quad x\in{\mathbb{R}^N}, $\end{document} where $ V $, $ K $ are smooth functions and $ V $ may vanish at infinity, $ 2<q<2(2^*) $. We prove the existence of a positive ground state solution which possesses a unique local maximum and decays exponentially.