Degree powers in $C_5$-free graphs

Let $G$ be a graph with degree sequence $d_1,d_2,\ldots,d_n$. Given a positive integer $p$, denote by $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given an integer $p$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order $r+1$ and a cycle of even length, respectively. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number. In this paper, we consider $ex_p(n, C_5)$ and prove that for any positive integer $p$ and sufficiently large $n$, there exists a constant $c=c(p)$ such that the following holds: if $ex_p(n, C_5)=e_p(G)$ for some $C_5$-free graph $G$ of order $n$, then $G$ is a complete bipartite graph having one vertex class of size $cn+o(n)$ and the other $(1-c)n+o(n)$.