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Article
On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems
IMA Journal of Applied Mathematics
  • Yi Li, The University of Iowa
  • Chunshan Zhao, Georgia Southern University
Document Type
Article
Publication Date
4-1-2007
DOI
10.1093/imamat/hxl032
Disciplines
Abstract

Abstract: We study the shape of least-energy solutions to the quasilinear elliptic equation ∊mΔmu − um−1 + f(u) = 0 with homogeneous Neumann boundary condition as ∊ → 0+ in a smooth bounded domain Ω ⊂ ℝN. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as ∊ → 0+, which plays an important role to locate the global maximum. Secondly, based on this sharp upper bound for the least energy, we show that the least-energy solutions concentrate on a point P∊ and dist(P∊, ∂Ω)/∊ goes to zero as ∊ → 0+. We also give an approximation result and find that as ∊ → 0+ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(∊) of P∊ where they concentrate.

Citation Information
Yi Li and Chunshan Zhao. "On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems" IMA Journal of Applied Mathematics Vol. 72 Iss. 2 (2007) p. 113 - 139 ISSN: 1464-3634
Available at: http://works.bepress.com/chunshan_zhao/42/