2024/03/25 by Franziska Borer, Borer, Franziska, Marcos T. O. Pimenta +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2403.17172
openalex publication_date 2024/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider degenerate Kirchhoff-type equations of the form -ϕ(Ξ(u)) (A(u)-|u|p-2u) = f(x,u) in Ω, \phantomaaiaaaaaaaaaϕ(Ξ(u)) B(u) ⋅ ν= g(x,u) on ∂Ω, where Ω⊆ ℝN, N≥ 2, is a bounded domain with Lipschitz boundary ∂Ω, A denotes the double phase operator given by A(u)=div (|∇ u|p-2∇ u + μ(x) |∇ u|q-2∇ u ) for u∈ W1,H(Ω), ν(x) is the outer unit normal of Ω at x ∈ ∂Ω, B(u)=|∇ u|p-2∇ u + μ(x) |∇ u|q-2∇ u, \phantomaaaiaaaaΞ(u)= ∫Ω((|∇ u|p+|u|p)/(p)+μ(x) (|∇ u|q)/(q)) d x, 10 and ζ≥ 1, and f\colonΩ×ℝ→ℝ, g\colon∂Ω×ℝ→ℝ are Carathéodory functions that grow superlinearly and subcritically. We prove the existence of a nodal ground state solution to the problem above, based on variational methods and minimization of the associated energy functional E\colon W1,H(Ω) →ℝ over the constraint set C=\u ∈ W1,H(Ω)\colon u±≠ 0, ⟨ E'(u),u+ ⟩= ⟨ E'(u),-u- ⟩=0 \, whereby C differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.