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

We study positive solutions of the equationΔu=u−p−1in Ω⊂RN(N⩾2), where p>0p>0 and Ω   is a bounded or unbounded domain. We show that there is a number pc=pc(N)⩾0pc=pc(N)⩾0 such that this equation with Ω=RNΩ=RN has no stable positive solution for p>pcp>pc. We further show that there is a critical power pc=pc(N)pc=pc(N) such that if p>pcp>pc, this equation with Ω=Br\{0}Ω=Br\{0} has no positive solution with finite Morse index that has an isolated rupture at 0; if 0max{pc,pc}p>max{pc,pc}.