| Summary: | This paper is concerned with the following quasilinear Schrödinger system in $\mathbb{R}^N$:
\begin{equation*}
\begin{cases}
-\varepsilon^2\Delta u+V_1(x)u-\varepsilon^2\Delta(u^{2})u=K_1(x)|u|^{22^*-2}u+h_1(x,u,v)u, \\
-\varepsilon^2\Delta v+V_2(x)v-\varepsilon^2\Delta(v^{2})v=K_2(x)|v|^{22^*-2}v+h_2(x,u,v)v, \\
\end{cases}
\end{equation*}
where $N\ge3$, $V_i(x)$ is a nonnegative potential, $K_i(x)$ is a bounded positive function, $i=1,2.$ $h_1(x,u,v)u$ and $h_2(x,u,v)v$ are superlinear but subcritical functions. Under some proper conditions, minimax methods are employed to establish the existence of standing wave solutions for this system provided that $\varepsilon$ is small enough, more precisely, for any $m\in\mathbb N$, it has $m$ pairs of solutions if $\varepsilon$ is small enough. And these solutions $(u_\varepsilon,v_\varepsilon)\to(0,0)$ in some Sobolev space as $\varepsilon \to 0$. Moreover, we establish the existence of positive solutions when $\varepsilon=1$. The system studied here can model some interaction phenomena in plasma physics.
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