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

On steady states of van der Waals force driven thin film equations