Double- and single-layered sign-changing solutions to a singularly perturbed elliptic equation concentrating at a single sphere. Issue 3 (4th March 2017)

• Record Type: Journal Article
• Authors: Clapp, Mónica; Manna, Bhakti Bhusan
• Is Part Of: Communications in partial differential equations. Volume 42:Issue 3(2017)
• Page Start: 474
• Page End: 490
• Publication Date: 2017-03-04
• Subjects: Asymptotic behavior of solutions -- concentration of solutions -- nodal solution -- singularly perturbed elliptic equation -- variational methods; 35B25 -- 35B40 (35J20 -- 35J25); Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2017.1279627
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We exhibit new concentration phenomena for the problem as 𝜀 →0, in domains of the form where m ≥1, n ≥3 and Θ is a bounded smooth domain in ℝ n with, 𝜀 >0, and . For particular choices of Θ we establish the existence of sign-changing solutions which concentrate on spheres that converge to a single m -dimensional sphere contained in the boundary of Ω, developing, either a positive and a negative layer, or a single sign-changing layer. In the first case, the asymptotic profile of each layer, in the transversal direction to its concentration sphere, is a rescaling of the positive or the negative ground state to the limit problem whereas, in the second case, it is a rescaling of a nonradial sign-changing solution to this last problem.