Asymptotic Behaviors of a Class of N-Laplacian Neumann Problems with Large Diffusion

We study asymptotic behaviors of positive solutions to the equation ε^N Δ_Nu-u^N-1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of ℝ^N (N ≥ 2) as ε → ∞. First, we study the subcritical case and show that there is a uniform upper bound independent of ε ∈ (0, ∞) for all positive solutions, and that for N ≥ 3 any positive solution goes to a constant in C^1,α sense as ε → ∞ under certain assumptions on f (see [W.-M. Ni, I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc. 297 (1986) 351-368] for the case N = 2). Second, we study critical case and show the existence of least-energy solutions. We also prove that for ε ∈ [1, ∞) there is a uniform upper bound independent of ε for the least-energy solutions. As ε → ∞, we show that for N = 2 any least-energy solution must be a constant for sufficiently large ε and for N ≥ 3 all least-energy solutions approach a constant in C^1,α sense.