Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations

ISBN-13: 9780821891636 / Angielski

Jaeyoung Byeon

Opis

The authors study the following singularly perturbed problem: - 2 ?u 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 .

Spis treści

• 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