Bulletin of the
Korean Mathematical Society BKMS

This paper is concerned with the quasilinear Schr\"{o}dinger system \begin{equation}\label{0.1} \left\{ \begin{aligned} &-\Delta u+a(x)u-\Delta(u^2)u=F_u(u,v)+h(x) \quad \; x\in \mathbb{R}^N,\\ &-\Delta v+b(x)v-\Delta(v^2)v=F_v(u,v)+g(x) \quad \; x\in \mathbb{R}^N, \end{aligned} \right. \end{equation} where $ N \geq 3$. The potential functions $a(x), b(x)\in L^\infty(\mathbb{R}^N)$ are bounded in $\mathbb{R}^N$. By using mountain pass theorem and the Ekeland variational principle, we prove that there are at least two solutions to system (\ref{0.1}).

Keywords: quasilinear Schr\"{o}dinger system, mountain pass theorem, Ekeland's variational principle

MSC numbers: 35J20, 35J70, 35J91