Extremal numbers of disjoint triangles in $r$-partite graphs

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by $ex(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining $ex(K_n,F)$ for a given graph $F$ is a classical extremal problem in graph theory. In 1962, Erd\H{o}s determined $ex(K_n,kK_3)$, which generalized Mantel's Theorem. On the other hand, in 1974, Bollob\H{o}s, Erd\H{o}s, and Straus determined $ex(K_{n_1,n_2,\dots,n_r},K_t)$, which extended Tur\'{a}n's Theorem to complete multipartite graphs. As a generalization of above results, in this paper, we determine $ex(K_{n_1,n_2,\dots,n_r},kK_3)$ for $r\ge 5$ and $15k\le n_1+4k\le n_2\le n_3\le \cdots \le n_r$.