On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems

Authors

Yi Li, Wright State University - Main CampusFollow
Chunshan ZhaoFollow

Document Type

Article

Publication Date

5-2005

Abstract

We study the structure of positive solutions to the equation ɛ^mΔ_mu-u^m-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ɛ→0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ɛ for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m⩽2 holds and ɛ is sufficiently large, any positive solution must be a constant.

Repository Citation

Li, Y., & Zhao, C. (2005). On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems. Journal of Differential Equations, 212 (1), 208-233.
https://corescholar.libraries.wright.edu/math/105

DOI

dx.doi.org/10.1016/j.jde.2004.07.021