Asymptotic behaviors of a class of [formula omitted]-Laplacian Neumann problems with large diffusion

Abstract

We study asymptotic behaviors of positive solutions to the equation ε N Δ N u − 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.