Normalized solutions for an horizontal transmission problem. Issue 15 (18th November 2021)

• Record Type: Journal Article
• Authors: Figueiredo, Giovany M.; Siciliano, Gaetano
• Is Part Of: Applicable analysis. Volume 100:Issue 15(2021)
• Page Start: 3174
• Page End: 3181
• Publication Date: 2021-11-18
• Subjects: Nonlinear transmission problem -- variational methods -- Krasnoselskii genus -- infinitely many solutions; 35J50 -- 35J57 -- 35D30; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1712371
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: Let Ω be a bounded and smooth domain in R N, N ≥ 3 . We study in this paper the following nonlinear transmission problem 1 − Δ u + λ u = | u | p − 2 u in Ω, − Δ v + μ v = a ( x ) | v | q − 2 v in R N ∖ Ω, u = v a n d ∂ u ∂ n = ∂ v ∂ n = 0 on ∂ Ω, where the unknowns are u : Ω → R, v : R N ∖ Ω → R and the real numbers λ, μ . Actually we are looking for solutions u, v with prescribed L 2 -norm. The problem has a variational formulation and indeed, under minimal assumptions, we are able to find infinitely many solutions by using the Krasnoselkii genus, whatever the L 2 − norm of u and v a priori is.