Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

In this paper we consider degenerate Kirchhoff-type equations of the form $$-\phi (\mathrm{\Xi }(u))(\mathcal{A}(u)-\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u)=f(x,u)\text{in }\mathrm{\Omega },$$ $$\phantom{aaiaaaaaaaaa}\phi (\mathrm{\Xi }(u))\mathcal{B}(u)\cdot \nu =g(x,u)\text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega }$, $\mathcal{A}$ denotes the double phase operator given by \begin{align*} $\mathcal{A}$(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u \right)\quad $\text{for }$u\in W^{1,$\mathcal{H}$}(Ω), \end{align*} $\nu (x)$ is the outer unit normal of $\mathrm{\Omega }$ at $x\in \mathrm{\partial }\mathrm{\Omega }$, $$\mathcal{B}(u)=\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+\mu (x)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u,$$ $$\phantom{aaaiaaaa}\mathrm{\Xi }(u)={\int }_{\mathrm{\Omega }}(\frac{\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\mathrm{\mid }u{\mathrm{\mid }}^p}{p}+\mu (x)\frac{\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q}{q})\mathrm{d}x,$$ $1<p<N$, $p<q<p^{\ast }=\frac{Np}{N-p}$, $0\le \mu (\cdot )\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $\phi (s)=a+b{s}^{\zeta -1}$ for $s\in \mathbb{R}$ with $a\ge 0$, $b>0$ and $\zeta \ge 1$, and $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$, $g\text{\hspace{0.17em}⁣}:\mathrm{\partial }\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ 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 $\mathcal{E}\text{\hspace{0.17em}⁣}:W^{1,\mathcal{H}}(\mathrm{\Omega })\to \mathbb{R}$ over the constraint set $$\mathcal{C}=\{u\in W^{1,\mathcal{H}}(\mathrm{\Omega })\text{\hspace{0.17em}⁣}:u^{\pm }\mathrm{\ne }0,⟨{\mathcal{E}}^{\mathrm{\prime }}(u),u^{+}⟩=⟨{\mathcal{E}}^{\mathrm{\prime }}(u),-u^{-}⟩=0\},$$ whereby $\mathcal{C}$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.