In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schr\"odinger system \begin{ equation*} \left\{ \begin{ array}{ ll} \varepsilon^{ 2s} (-\Delta)^{ s}u+V(x) u=Q_{ u}(u, v)+\frac{ 1}{ 2^{ *}_{ s}}K_{ u}(u, v) &\mbox{ in } \mathbb{ R}^{ N}\varepsilon^{ 2s} (-\Delta)^{ s}u+W(x) v=Q_{ v}(u, v)+\frac{ 1}{ 2^{ *}_{ s}}K_{ v}(u, v) &\mbox{ in } \mathbb{ R}^{ N} u, v>0 &\mbox{ in } \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 operator, $V:\mathbb{ R}^{ N}\rightarrow \mathbb{ R}$ and $W:\mathbb{ R}^{ N}\rightarrow \mathbb{ R}$ are positive H\"older continuous potentials, $Q$ and $K$ are homogeneous $C^{ 2}$-functions having subcritical and critical growth respectively. We relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.
Categories: math.AP