Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents. (July 2022)

• Record Type: Journal Article
• Authors: Jin, Sangdon
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: 35J60 -- 35J20 -- 58E05; Expanding domain -- Critical problem -- Multi-bump solutions; 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.2022.112829
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the existence of multi-bump solutions of − Δ u − λ u = u p, u > 0 in Ω t, u = 0 on ∂ Ω t, where λ is a real constant, N ≥ 3, p = N + 2 N − 2, and Ω t is a domain that expands as t → ∞ . It is known that the topology of the domain Ω t affects the existence and non-existence of solutions. In this article, we prove the existence of multi-bump solutions for more general domains; for example, Ω t b a l l ≡ { t q + v | q ∈ M, v ∈ ( T q M ) ⊥, | v | < 1 } or Ω t a n n = { t q + v | q ∈ M, v ∈ ( T q M ) ⊥, a < | v | < a + 1 }, where a > 0 and M is a k -dimensional compact smooth submanifold of R N without boundary ( k = 1, ⋯, N − 1 ) .