## Abstract

We study the shape of least-energy solutions to the quasilinear elliptic equation ε^{m} Δ_{m}u - u^{m-1} + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0^{+} in a smooth bounded domain Ω ⊂ ℝ^{N}. 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 P_{ε} and dist (P_{ε} ∂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 P_{ε} where they concentrate.

Original language | American English |
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Journal | IMA Journal of Applied Mathematics |

Volume | 72 |

DOIs | |

State | Published - Apr 1 2007 |

## Keywords

- Least-Energy Solution
- M-Laplacian Operator
- Quasi-Linear Neumann Problem

## DC Disciplines

- Education
- Mathematics