Uniqueness of single-peaked solutions to singularity perturbed semilinear Dirichlet problems. (October 2021)

• Record Type: Journal Article
• Authors: Tian, Shuying; Zhan, Jinpeng
• Is Part Of: Applied mathematics letters. Volume 120(2021)
• Publication Date: 2021-10
• Subjects: Single-peaked solutions -- Degeneracy -- Uniqueness -- Pohozaev identity; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2021.107338
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: We are concerned with the uniqueness of single-peaked solutions to the following problem − ε 2 △ u + u = Q ( x ) u p − 1, in Ω, u > 0, in Ω, u = 0, on ∂ Ω, where ε > 0 is a small parameter, and N ≥ 3, 2 < p < 2 N ∕ ( N − 2 ) . By local Pohozaev identity and blow-up analysis, we show the uniqueness of single-peaked solutions under certain assumptions on asymptotic behavior of Q ( x ) and its first derivatives near the critical point. Here the degeneracy of the critical point is also allowed, which gives a partial answer to the question proposed in Cao et al. (1998).