Fučik spectrum for the Kirchhoff-type problem and applications. (May 2019)

• Record Type: Journal Article
• Authors: Li, Fuyi; Rong, Ting; Liang, Zhanping
• Is Part Of: Nonlinear analysis. Volume 182(2019)
• Page Start: 280
• Page End: 302
• Publication Date: 2019-05
• Subjects: Fučik spectrum -- Kirchhoff-type problem -- Multiplicity of 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.2018.12.021
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this study, we focus on the Fučik spectrum for the Kirchhoff-type problem, which is defined as a set Σ comprising those ( α, β ) ∈ R 2 such that (0.1) − ∫ Ω | ∇ u | 2 Δ u = α ( u + ) 3 + β ( u − ) 3, in Ω, u = 0, on ∂ Ω has a nontrivial solution, where Ω is an open ball in R N for N = 1, 2, 3 ; or Ω ⊂ R 2 is symmetric in x and y, and convex in the x and y directions, u + = max { u, 0 }, u − = min { u, 0 }, and u = u + + u − . First, we prove that the curves { μ 1 } × R, R × { μ 1 }, and C ≔ { ( s + c ( s ), c ( s ) ) : s ∈ R } belong to Σ, where c ( s ) = min { β : ( s + β, β ) ∈ Σ 0 } and Σ 0 comprises those ( α, β ) ∈ R 2 such that(0.1) has a sign changing solution. We refer to { μ 1 } × R and R × { μ 1 } as trivial curves in Σ in the sense that any solution of(0.1) with ( α, β ) ∈ { μ 1 } × R or R × { μ 1 } is signed. We denote C as the first nontrivial curve in Σ in the sense that any solution of(0.1) with ( α, β ) ∈ C is sign changing and for each s ∈ R, we consider the line that passes through ( s, 0 ) with a slope of 1 in the α O β plane R 2, then the first point on this line that intersects with Σ 0 is simply ( s + c ( s ), c ( s ) ) ∈ C . Second, we investigate some properties of the function c and the curve C . In particular, c is Lipschitz continuous, decreasing on R and c ( s ) → μ 1 as s → ∞, and C is asymptotic to the broken line ℒ 2 ≔ { μ 1 } × [ μ 1, ∞ ) ∪ [ μ 1, ∞ ) × { μ 1 } . Furthermore, we show that the point ( α, β ) corresponding to the signed solution of(0.1) is from ℒ ≔ ( { μ 1 } × R ) ∪ ( R × { μ 1 } ), the point ( α, β ) corresponding to the sign changing solution of(0.1) is on the upper right of ℒ 2, and no nontrivial solution of(0.1) exists when ( α, β ) is between ℒ 2 and C . Finally, as an application, we establish the multiplicity of solutions to the following Kirchhoff-type problem: − 1 + ∫ Ω | ∇ u | 2 Δ u = f ( x, u ), in Ω, u = 0, on ∂ Ω, where the nonlinearity f is asymptotically linear at zero and asymptotically 3-linear at infinity. To the best of our knowledge, this is the first study to consider that the nonlinearity has an extension property at both the zero and infinity points.