Multiplicity and concentration of solutions for fractional Schr\"odinger systems via penalization method

The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}, \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v) &\mbox{ in } \mathbb{R}^{N}, u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth. In order to relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values, we apply penalization techniques, Nehari manifold arguments and Ljusternik-Schnirelmann theory.