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 m Δmu − u m−1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of (N ≥ 2). First, we study subcritical case for 2 \u3c m \u3c N and show that after passing by a sequence positive solutions go to a constant in C1, α sense as ε → ∞. Second, we study the critical case for 1 \u3c m \u3c 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 \u3c m ≤ 2 the least energy solutions must be a constant if ε is sufficiently large and for 2 \u3c m \u3c N the least energy solutions go to a constant in C1, α sense as ε → ∞