# 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 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. 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 C^{1,α} sense.

## Recommended Citation

Zhao, Chunshan.
2008.
"Asymptotic Behaviors of a Class of *N*-Laplacian Neumann Problems with Large Diffusion."
*Nonlinear Analysis: Theory, Methods & Applications*, 69 (8): 2496-2524.
doi: 10.1016/j.na.2007.08.028

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/250