Let Ω ⊂ ℝN, N ≥ 2 be a bounded smooth domain and α > 1. We are interested in the singular elliptic equation Δh = 1/αh-α - p in Ω with Neumann boundary conditions. In this paper, a complete description of all continuous radially symmetric solutions is given. In particular, we construct 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 α = 3, the equation is used to model steady states of van der Waals force driven thin films of viscous fluids. We also consider the physical problem when total volume of the fluid is prescribed.
Bibliographical noteFunding Information:
We wish to thank Richard Laugesen, Mary Pugh and Dejan Slepcˇev and two anonymous referees for helpful comments. In particular, we wish to thank Mary Pugh for pointing out a related work of Bertozzi et al. . The research is supported in part by the NSF.