Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations

Author(s)

    • Byeon, Jaeyoung
    • Tanaka, Kazunaga

Bibliographic Information

Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations

Jaeyoung Byeon, Kazunaga Tanaka

(Memoirs of the American Mathematical Society, v. 229, no. 1076)

American Mathematical Society, 2014

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Note

"Volume 229, Number 1076 (third of 5 numbers), May 2014"

Bibliography: p. 87-89

Description and Table of Contents

Description

The authors study the following singularly perturbed problem: −ϵ 2 Δu V(x)u=f(u) in R N . Their main result is the existence of a family of solutions with peaks that cluster near a local maximum of V(x) . A local variational and deformation argument in an infinite dimensional space is developed to establish the existence of such a family for a general class of nonlinearities f .

Table of Contents

Introduction and results Preliminaries Local centers of mass Neighborhood Ω ϵ (ρ,R,β) and minimization for a tail of u in Ω ϵ A gradient estimate for the energy functional Translation flow associated to a gradient flow of V(x) on R N Iteration procedure for the gradient flow and the translation flow An (N 1)ℓ 0 -dimensional initial path and an intersection result Completion of the proof of Theorem 1.3 Proof of Proposition 8.3 Proof of Lemma 6.1 Generalization to a saddle point setting Bibliography

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