Fučik spectrum for the Kirchhoff-type problem and applications. (May 2019) - Li, Fuyi; Rong, Ting; Liang, Zhanping
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.