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This paper is concerned with a quasilinear Schr\"{o}dinger system in $\mathbbR^{N}$ $$\left\{\aligned &-\Deltau+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\&-\Deltav+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\& u(x)\to 0\ \hbox{and}\quad v(x)\to 0\ \hbox{as}\ |x|\to\infty,\endaligned\right. $$ where $\alpha,\beta>1$ and$2<\alpha+\beta<\frac{4N}{N-2}$ ($N \geq 3$). $A(x)$ and $B(x)$ are twoperiodic functions. By minimization under a convenient constraint andconcentration-compactness lemma, we prove the existence of ground statessolution. Our result covers the case of $\alpha+\beta\in(2,4)$ which seems tobe the first result for coupled quasilinear Schr\"{o}dinger system in theperiodic situation.

Ground states solution of Nehari-Poho\v{z}aev type for periodic quasilinear Schr\"{o}dinger system