## 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 |