### Abstract

We consider the following singularly perturbed Neumann problem: ε^{2} Δu - u + f(u) = 0 in Ω, ∂u/∂u = 0 on ∂Ω, where Δ = Σ_{i=1}^{N} ∂^{2}/∂_{i}^{2} is the Laplace operator, ε > 0 is a constant, Ω is a bounded, smooth domain in ℝ^{N} with its unit outward normal v, and f is superlinear and subcritical. A typical f is f (u) = u^{p} where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N - 2) when N ≥ 3. We show that there exists an ε_{0} > 0 such that for 0 < ε < ε_{0} and for each integer K bounded by 1 ≤ K ≤ ^{α}N,Ω,f/ε^{N}(|ln ε|^{N} where ^{α}N,Ω,f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for ^{α}N,Ω,f is also given.) As a consequence, we obtain that for ε sufficiently small, there exists at least [^{α}N, Ω,f/ε^{N}(|ln ε|^{N}] number of solutions. Moreover, for each m ε (0, N) there exist solutions with energies in the order ε^{N-m}.

Original language | English (US) |
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Pages (from-to) | 252-281 |

Number of pages | 30 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*60*(2), 252-281. https://doi.org/10.1002/cpa.20139

**On the number of interior peak solutions for a singularly perturbed neumann problem.** / Lin, Fang-Hua; Ni, Wei Ming; Wei, Jun Cheng.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 60, no. 2, pp. 252-281. https://doi.org/10.1002/cpa.20139

}

TY - JOUR

T1 - On the number of interior peak solutions for a singularly perturbed neumann problem

AU - Lin, Fang-Hua

AU - Ni, Wei Ming

AU - Wei, Jun Cheng

PY - 2007/2

Y1 - 2007/2

N2 - We consider the following singularly perturbed Neumann problem: ε2 Δu - u + f(u) = 0 in Ω, ∂u/∂u = 0 on ∂Ω, where Δ = Σi=1N ∂2/∂i2 is the Laplace operator, ε > 0 is a constant, Ω is a bounded, smooth domain in ℝN with its unit outward normal v, and f is superlinear and subcritical. A typical f is f (u) = up where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N - 2) when N ≥ 3. We show that there exists an ε0 > 0 such that for 0 < ε < ε0 and for each integer K bounded by 1 ≤ K ≤ αN,Ω,f/εN(|ln ε|N where αN,Ω,f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for αN,Ω,f is also given.) As a consequence, we obtain that for ε sufficiently small, there exists at least [αN, Ω,f/εN(|ln ε|N] number of solutions. Moreover, for each m ε (0, N) there exist solutions with energies in the order εN-m.

AB - We consider the following singularly perturbed Neumann problem: ε2 Δu - u + f(u) = 0 in Ω, ∂u/∂u = 0 on ∂Ω, where Δ = Σi=1N ∂2/∂i2 is the Laplace operator, ε > 0 is a constant, Ω is a bounded, smooth domain in ℝN with its unit outward normal v, and f is superlinear and subcritical. A typical f is f (u) = up where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N - 2) when N ≥ 3. We show that there exists an ε0 > 0 such that for 0 < ε < ε0 and for each integer K bounded by 1 ≤ K ≤ αN,Ω,f/εN(|ln ε|N where αN,Ω,f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for αN,Ω,f is also given.) As a consequence, we obtain that for ε sufficiently small, there exists at least [αN, Ω,f/εN(|ln ε|N] number of solutions. Moreover, for each m ε (0, N) there exist solutions with energies in the order εN-m.

UR - http://www.scopus.com/inward/record.url?scp=33846284180&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846284180&partnerID=8YFLogxK

U2 - 10.1002/cpa.20139

DO - 10.1002/cpa.20139

M3 - Article

VL - 60

SP - 252

EP - 281

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 2

ER -