Generating loops and isolas in semilinear elliptic BVP's. (July 2023)

• Record Type: Journal Article
• Authors: López-Gómez, Julián; Sampedro, Juan Carlos
• Is Part Of: Nonlinear analysis. Volume 232(2023)
• Publication Date: 2023-07
• Subjects: 35B32 -- 35B09 -- 35B45; Loops and isolas -- Positive and negative solutions -- Bi-parametric bifurcation theory for Fredholm operators; 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.2023.113268
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we ascertain the global λ -structure of the set of positive and negative solutions bifurcating from u = 0 for the semilinear elliptic BVP − d Δ u = λ 〈 a, ∇ u 〉 + u + λ u 2 − u q in Ω, u = 0 on ∂ Ω, according to the values of d > 0 and the integer number q ≥ 4 . Figs. 1.1–1.3 summarize the main findings of this paper according to the values of d and q . Note that the role played by the parameter λ in this model is very special, because, besides measuring the strength of the convection, it quantifies the amplitude of the nonlinear term λ u 2 . We regard to this problem as a mathematical toy to generate solution loops and isolas in Reaction Diffusion equations.