We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-��_p)^r u = ��\dfrac��p|u|^{��-2}u|v|^�� &\text{in } ��,\vspace{.1cm} (-��_p)^s u = ��\dfrac��p|u|^��|v|^{��-2}v &\text{in } ��, u=v=0 &\text{in }��^c=\R^N\setminus��. \end{cases} $$ We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $��_n$ such that $��_n\to\infty$ as $n\to\infty$. In addition, we study the limit as $p\to \infty$ of the first eigenvalue, $��_{1,p}$, and we obtain $ [��_{1,p}]^{\nicefrac{1}{p}}\to ��_{1,\infty} $ as $p\to\infty,$ where $$ ��_{1,\infty} = \inf_{(u,v)} \left\{ \frac{\max \{ [u]_{r,\infty} ; [v]_{s,\infty} \} }{ \| |u|^�� |v|^{1-��} \|_{L^\infty (��)} } \right\} = \left[ \frac{1}{R(��)} \right]^{ (1-��) s + ��r }. $$ Here $R(��):=\max_{x\in��}\dist(x,\partial��)$ and $[w]_{t,\infty} \coloneqq \sup_{(x,y)\in\overline��} \frac{| w(y) - w(x)|}{|x-y|^{t}}.$ Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigen-pairs.

19 pages