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 .
Jaeyoung Byeon, KAIST, Daejeon, Republic of Korea.Kazunaga Tanaka, Waseda University, Tokyo, Japan.
Introduction and resultsPreliminariesLocal centers of massNeighborhood Ω ϵ (ρ,R,β) and minimization for a tail of u in Ω ϵA gradient estimate for the energy functionalTranslation flow associated to a gradient flow of V(x) on R NIteration procedure for the gradient flow and the translation flowAn (N 1)ℓ 0 -dimensional initial path and an intersection resultCompletion of the proof of Theorem 1.3Proof of Proposition 8.3Proof of Lemma 6.1Generalization to a saddle point settingBibliography