Ground states solution of Nehari-Pohožaev type for periodic quasilinear Schrödinger system

This paper is concerned with a quasilinear Schrödinger system in ${\mathbb{R}}^N$ \left\{\aligned &-Δu+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2α}{α+β}|u|^{α-2}u|v|^β,\\ &-Δv+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2β}{α+β}|u|^α|v|^{β-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 $2<\alpha +\beta <\frac{4N}{N-2}$ ($N\ge 3$). $A(x)$ and $B(x)$ are two periodic functions. By minimization under a convenient constraint and concentration-compactness lemma, we prove the existence of ground states solution. Our result covers the case of $\alpha +\beta \in (2,4)$ which seems to be the first result for coupled quasilinear Schrödinger system in the periodic situation.