Degenerate Kirchhoff problems with nonlinear Neumann boundary condition
Franziska Borer, Marcos T. O. Pimenta, Patrick Winkert
TL;DR
This work addresses a degenerate Kirchhoff-type PDE with a nonlinear Neumann boundary condition in a Musielak–Orlicz setting, incorporating a double phase operator and a nonlocal Kirchhoff term. The authors develop a variational framework with the energy functional $\mathcal{E}$ and constrain minimization on the nodal set $\mathcal{C}=\{u: u^{\pm}\neq 0, \langle \mathcal{E}'(u),u^+\rangle=\langle \mathcal{E}'(u),-u^-\rangle=0\}$ to obtain a least energy sign-changing solution. They establish the necessary functional-analytic groundwork in the Musielak–Orlicz space $W^{1,\mathcal{H}}(\Omega)$, including the properties of the operator $B$, embeddings, and deformation tools, and then prove the existence of a nodal ground state by showing the infimum of $\mathcal{E}$ on $\mathcal{C}$ is achieved at a critical point. The results extend Kirchhoff-type analyses to nonstandard growth with degenerate tension ($a\ge 0$) and nonlinear Neumann data, providing a rigorous existence theory for physically relevant models with nonlocal and boundary nonlinearities.
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 $Ω\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partialΩ$, $\mathcal{A}$ 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*} $ν(x)$ is the outer unit normal of $Ω$ at $x \in \partialΩ$, \[\mathcal{B}(u)=|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u,\] \[\phantom{aaaiaaaa}Ξ(u)= \int_Ω\left(\frac{|\nabla u|^p+|u|^p}{p}+μ(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x,\] $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq μ(\cdot)\in L^\infty(Ω)$, $φ(s) = a + b s^{ζ-1}$ for $s\in\mathbb{R}$ with $a \geq 0$, $b>0$ and $ζ\geq 1$, and $f\colonΩ\times\mathbb{R}\to\mathbb{R}$, $g\colon\partialΩ\times\mathbb{R}\to\mathbb{R}$ 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 $\mathcal{E}\colon W^{1,\mathcal{H}}(Ω) \to\mathbb{R}$ 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.