On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems
IMA Journal of Applied Mathematics
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.
Li, Yi, Chunshan Zhao.
"On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems."
IMA Journal of Applied Mathematics, 72 (2): 113-139.
doi: 10.1093/imamat/hxl032 source: https://academic.oup.com/imamat/article/72/2/113/663155