Fučik spectrum for the Kirchhoff-type problem and applications.

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. [ABSTRACT FROM AUTHOR]