TY - JOUR
T1 - On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems
AU - Li, Yi
AU - Zhao, Chunshan
PY - 2007/4/1
Y1 - 2007/4/1
N2 - We study the shape of least-energy solutions to the quasilinear elliptic equation εm Δmu - um-1 + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0+ in a smooth bounded domain Ω ⊂ ℝN. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as ε → 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 Pε and dist (Pε ∂Ω)/ε goes to zero as ε → 0+. We also give an approximation result and find that as ε → 0+ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(ε) of Pε where they concentrate.
AB - We study the shape of least-energy solutions to the quasilinear elliptic equation εm Δmu - um-1 + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0+ in a smooth bounded domain Ω ⊂ ℝN. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as ε → 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 Pε and dist (Pε ∂Ω)/ε goes to zero as ε → 0+. We also give an approximation result and find that as ε → 0+ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(ε) of Pε where they concentrate.
KW - Least-Energy Solution
KW - M-Laplacian Operator
KW - Quasi-Linear Neumann Problem
UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/689
UR - https://doi.org/10.1093/imamat/hxl032
U2 - 10.1093/imamat/hxl032
DO - 10.1093/imamat/hxl032
M3 - Article
SN - 0272-4960
VL - 72
JO - IMA Journal of Applied Mathematics
JF - IMA Journal of Applied Mathematics
ER -