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

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208-233, 2005) to study the structure of positive solutions to the equation ε^mΔ_mu - u^m-1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of ℝ^N (N ≥ 2). First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C^1,α sense as ε → ∞. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of ε ∈ [1,∞) for the least-energy solutions. Third, we show that in the critical case for 1 < m ≤ 2 the least energy solutions must be a constant if ε is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C^1,α sense as ε → ∞.