## Abstract

We study asymptotic behaviors of positive solutions to the equation ε^{N} Δ_{N}u-u^{N-1} + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of ℝ^{N} (N ≥ 2) as ε → ∞. First, we study the subcritical case and show that there is a uniform upper bound independent of ε ∈ (0, ∞) for all positive solutions, and that for N ≥ 3 any positive solution goes to a constant in C^{1,α} sense as ε → ∞ under certain assumptions on f (see [W.-M. Ni, I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc. 297 (1986) 351-368] for the case N = 2). Second, we study critical case and show the existence of least-energy solutions. We also prove that for ε ∈ [1, ∞) there is a uniform upper bound independent of ε for the least-energy solutions. As ε → ∞, we show that for N = 2 any least-energy solution must be a constant for sufficiently large ε and for N ≥ 3 all least-energy solutions approach a constant in C^{1,α} sense.

Original language | American English |
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Journal | Nonlinear Analysis: Theory, Methods & Applications |

Volume | 69 |

DOIs | |

State | Published - Oct 15 2008 |

## Keywords

- Asymptotic behavior
- Harnack inequality
- Least-energy solution
- N-Laplacian
- Quasilinear Neumann problem

## DC Disciplines

- Education
- Mathematics

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