  Let $R$ be a polynomial ring over a field $K$ of arbitrary characteristic and
$D$ be the ring of differential operators over $R$. Inspired by Euler formula
for homogeneous polynomials, we introduce a class of graded $D$-modules, called
Eulerian graded $D$-modules. It is proved that a vast class of $D$-modules,
including all composite of local cohomology modules,
$H_{J_0}^{i_0}(H_{J_1}^{i_1}...(H_{J_n}^{i_n}(R)))$ where $J_1,...,J_n$ are
homogeneous ideals of $R$, are Eulerian graded. As an application of our
theory, we prove that in all characteristic, these composite of local
cohomology modules must be isomorphic to a direct sum of $^{*}E(n)$, the graded
injective hull of $R/m$ shifted by $n$. This answers a question raised in
arXiv:1102.5336. An application of our theory of Eulerian graded $D$-modules to
the graded injective hull of $R/P$, where $P$ is a homogeneous prime ideal of
$R$, is discussed as well.
