Sign changing bubbling solutions for a critical Neumann problem in 3D. (November 2019)

• Record Type: Journal Article
• Authors: Salazar, Dora
• Is Part Of: Nonlinear analysis. Volume 188(2019)
• Page Start: 500
• Page End: 539
• Publication Date: 2019-11
• Subjects: Critical problem -- Neumann boundary condition -- Multi-spike solutions -- Sign changing; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.06.018
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let Ω be a smooth bounded domain in R 3 . We consider the problem Δ u − λ u + | u | 4 u = 0 in Ω, with zero Neumann boundary conditions, and investigate the existence of sign changing solutions, which concentrate simultaneously around k different points of Ω as λ approaches a special value λ 0 . We characterize the number λ 0 in terms of the Green function of − Δ + λ . In particular, we construct a family of solutions exhibiting bubbling behavior at two different points of an annular domain as λ tends to λ 0 . In doing so, we also derive a representation formula of the Green function of − Δ with Neumann boundary conditions in the annulus in terms of zonal harmonics.