On the structure of solutions to a class of quasilinear elliptic Neumann problems

We study the structure of positive solutions to the equation ɛmΔmu-um-1+fu=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ɛ→0+ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ɛ for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1