  An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$
of $G$ satisfies $\lambda (G') < \lambda (G)$, where $\lambda (G)$ is the
Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $\pi(F)$, the
Tur\'an density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the
supremum of the Lagrangians of all dense $F$-hom-free $r$-uniform hypergraphs.
This connection has been applied in estimating Tur\'an density of hypergraphs.
When $r=2$, the result of Motzkin and Straus shows that a graph is dense if and
only if it is a complete graph. However, when $r\ge 3$, it becomes much harder
to estimate the Lagrangians of $r$-uniform hypergraphs and to characterize the
structure of all dense $r$-uniform graphs. The main goal of this note is to
give some sufficient conditions for $3$-uniform graphs with given substructures
to be dense. For example, if $G$ is a $3$-graph with vertex set $[t]$ and $m$
edges containing $[t-1]^{(3)}$, then $G$ is dense if and only if $m \ge {t-1
\choose 3}+{t-2 \choose 2}+1$. We also give sufficient condition condition on
the number of edges for a $3$-uniform hypergraph containing a large clique
minus $1$ or $2$ edges to be dense.
