Finite Morse index steady states of van der Waals force driven thin film equations

We study positive solutions of the equation Δ u = u − p − 1 in Ω ⊂ R N ( N ⩾ 2 ) , where p > 0 and Ω is a bounded or unbounded domain. We show that there is a number p c = p c ( N ) ⩾ 0 such that this equation with Ω = R N has no stable positive solution for p > p c . We further show that there is a critical power p c = p c ( N ) such that if p > p c , this equation with Ω = B r \ { 0 } has no positive solution with finite Morse index that has an isolated rupture at 0; if 0 < p ⩽ p c , this equation with Ω = B r \ { 0 } has a positive solution with finite Morse index that has an isolated rupture at 0, and it has no positive solution with finite Morse index when Ω = R N \ B R provided p > max { p c , p c } .