  Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein
ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local
cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i
\not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to
$R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only
if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated
singularity resp. if $I$ is set-theoretically a complete intersection in
codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$
for all $i > m, m$ a given integer, resp. an estimate of the dimension of
$H^i_I(R).$
