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

We study the structure of positive solutions to the equation ε^mΔ_mu -u^m-1 + f(u) = 0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ε → 0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ε for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1 < m ≤ 2 holds and ε is sufficiently large, any positive solution must be a constant.