TY - JOUR
T1 - Locating the Peaks of Least-Energy Solutions to a Quasi-Linear Elliptic Neumann Problem, Part II
AU - Zhao, Chunshan
PY - 2010/6/1
Y1 - 2010/6/1
N2 - We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to study the shape of least-energy solutions to the quasilinear problem εm Δm u - um - 1 + f (u) = 0 with homogeneous Neumann boundary condition. In this paper we focus on the case 1 < m < 2 as a complement to our previous work on the case m ≥ 2. We use an intrinsic variation method to show that as the case m ≥ 2, when ε → 0+, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂ Ω at a rate of o (ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂ Ω.
AB - We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to study the shape of least-energy solutions to the quasilinear problem εm Δm u - um - 1 + f (u) = 0 with homogeneous Neumann boundary condition. In this paper we focus on the case 1 < m < 2 as a complement to our previous work on the case m ≥ 2. We use an intrinsic variation method to show that as the case m ≥ 2, when ε → 0+, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂ Ω at a rate of o (ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂ Ω.
KW - Exponential decay
KW - Least-energy solution
KW - Mean curvature
KW - Quasilinear Neumann problem
KW - m-Laplacian operator
UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/246
UR - https://doi.org/10.1016/j.na.2010.01.049
U2 - 10.1016/j.na.2010.01.049
DO - 10.1016/j.na.2010.01.049
M3 - Article
VL - 72
JO - Nonlinear Analysis: Theory, Methods & Applications
JF - Nonlinear Analysis: Theory, Methods & Applications
ER -