  Let $\Omega\subset\mathbb{R}^{N}$, $N\geq2$ be a bounded smooth domain and
$\alpha>1$. We are interested in the singular elliptic equation% \[ \triangle
h=\frac{1}{\alpha}h^{-\alpha}-p\quad\text{in}\Omega \] with Neumann boundary
conditions. In this paper, we gave a complete description of all continuous
radially symmetric solutions. In particular, we constructed nontrivial smooth
solutions as well as rupture solutions. Here a continuous solution is said to
be a rupture solution if its zero set is nonempty. When N=2 and $\alpha=3$, the
equation has been used to model steady states of van der Waals force driven
thin films of viscous fluids. We also considered the physical problem when
total volume of the fluid is prescribed.
