  In this paper, we study the existence and non-existence result of positive
solutions to a singular elliptic equation with negative power on the bounded
smooth domain or in the whole Euclidean space. Our model arises in the study of
the steady states of thin films and other applied physics. We can get some
useful local gradient estimate and L1 lower bound for positive solutions of the
elliptic equation. A uniform positive lower bound for convex positive solutions
is also obtained. We show that in lower dimensions, there is no stable positive
solutions in the whole space. In the whole space of dimension two, we can show
that there is no positive smooth solution with finite Morse index. Symmetry
properties of related integral equations are also given.
