Mathematical Sciences: Faculty Publications
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.
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
Copyright
This work is archived and distributed under the repository's Standard Copyright and Reuse License (opens in new tab). End users may copy, store, and distribute this work without restriction. For all other uses, permission must be obtained from the copyright owners or their authorized agents.
