On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems

Document Type

Article

Publication Date

4-2007

Publication Title

IMA Journal of Applied Mathematics

DOI

10.1093/imamat/hxl032

ISSN

1464-3634

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.

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