Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

Abstract

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,ϕ(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, [Formula presented], 0≤μ(⋅)∈L∞(Ω), ϕ(s)=a+bsζ−1 for s∈R with a≥0, b>0 and ζ≥1, and f:Ω×R→R, g:∂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:W1,H(Ω)→R over the constraint set C={u∈W1,H(Ω):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.