Asymptotic behaviors of a class of NN-Laplacian Neumann problems with large diffusion

We study asymptotic behaviors of positive solutions to the equation εNΔNuεNΔNu−uN−1+f(u)=0−uN−1+f(u)=0 with homogeneous Neumann boundary condition in a smooth bounded domain of RNRN(N≥2)(N≥2) as ε→∞ε→∞. First, we study the subcritical case and show that there is a uniform upper bound independent of ε∈(0,∞)ε∈(0,∞) for all positive solutions, and that for N≥3N≥3 any positive solution goes to a constant in C1,αC1,α sense as ε→∞ε→∞ under certain assumptions on ff (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=2N=2). Second, we study critical case and show the existence of least-energy solutions. We also prove that for ε∈[1,∞)ε∈[1,∞) there is a uniform upper bound independent of εε for the least-energy solutions. As ε→∞ε→∞, we show that for N=2N=2 any least-energy solution must be a constant for sufficiently large εε and for N≥3N≥3 all least-energy solutions approach a constant in C1,αC1,α sense.