Title

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

Document Type

Article

Publication Date

10-15-2008

Publication Title

Nonlinear Analysis: Theory, Methods & Applications

DOI

10.1016/j.na.2007.08.028

ISSN

0362-546X

Abstract

We study asymptotic behaviors of positive solutions to the equation εNΔNu−uN−1+f(u)=0 with homogeneous Neumann boundary condition in a smooth bounded domain of RN(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 C1,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. Asε→∞, we show that for N=2 any least-energy solution must be a constant for sufficiently large ε and forN≥3 all least-energy solutions approach a constant in C1,α sense.

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