The generalized Turán number of $2S_{\ell }$

The generalized Turán number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $S_{\ell }$ denote the star on $\ell +1$ vertices, and let $k{S}_{\ell }$ denote the disjoint union of $k$ copies of $S_{\ell }$. Gan et al. and Chase determined $ex(n,K_s,S_{\ell })$ for all integers $s\ge 3$, $\ell \ge 1$ and $n\ge 1$. In this paper, we determine $ex(n,K_s,2S_{\ell })$ for all integers $s\ge 4$, $\ell \ge 1$ and $n\ge 1$.