Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

Abstract

In this paper we consider degenerate Kirchhoff-type equations of the form \[-ϕ(Ξ(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } Ω,\] \[\phantom{aaiaaaaaaaaa}ϕ(Ξ(u)) \mathcal{B}(u) \cdot ν= g(x,u) \quad \text{on } \partialΩ,\] where $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary $\partial \hat{I}\text{c◯}$, $\mathcal{A}$ 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*} $\hat{I}½(x)$ is the outer unit normal of $\hat{I}\text{c◯}$ at $x\in \partial \hat{I}\text{c◯}$, \[\mathcal{B}(u)=|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u,\] \[\phantom{aaaiaaaa}Ξ(u)= \int_Ω\left(\frac{|\nabla u|^p+|u|^p}{p}+μ(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x,\] $1<p<N$, $p<q<p^{\ast }=\frac{Np}{N-p}$, $0\le \hat{I}¼(\cdot )\in L^{\infty }(\hat{I}\text{c◯})$, $\ddot{I}•(s)=a+b{s}^{\hat{I}\P -1}$ for $s\in \mathbb{R}$ with $a\ge 0$, $b>0$ and $\hat{I}\P \ge 1$, and $f:\hat{I}\text{c◯}\times \mathbb{R}\to \mathbb{R}$, $g:\partial \hat{I}\text{c◯}\times \mathbb{R}\to \mathbb{R}$ 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 $\mathcal{E}:W^{1,\mathcal{H}}(\hat{I}\text{c◯})\to \mathbb{R}$ over the constraint set \[\mathcal{C}=\Big\{u \in W^{1,\mathcal{H}}(Ω)\colon u^{\pm}\neq 0,\, \left\langle \mathcal{E}'(u),u^+ \right\rangle= \left\langle \mathcal{E}'(u),-u^- \right\rangle=0 \Big\},\] whereby $\mathcal{C}$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.