Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

Abstract

In this paper we consider degenerate Kirchhoff-type equations of the form \[-\phi(\Xi(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } \Omega,\] \[\phantom{aaiaaaaaaaaa}\phi (\Xi(u)) \mathcal{B}(u) \cdot \nu = g(x,u) \quad \text{on } \partial\Omega,\] where $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $\mathcal{A}$ denotes the double phase operator given by \begin{align*} \mathcal{A}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u \right)\quad \text{for }u\in W^{1,\mathcal{H}}(\Omega), \end{align*} $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, \[\mathcal{B}(u)=|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u,\] \[\phantom{aaaiaaaa}\Xi(u)= \int_\Omega \left(\frac{|\nabla u|^p+|u|^p}{p}+\mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x,\] $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq \mu(\cdot)\in L^\infty(\Omega)$, $\phi(s) = a + b s^{\zeta-1}$ for $s\in\mathbb{R}$ with $a \geq 0$, $b>0$ and $\zeta \geq 1$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$, $g\colon\partial\Omega\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory 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 $\mathcal{E}\colon W^{1,\mathcal{H}}(\Omega) \to\mathbb{R}$ over the constraint set \[\mathcal{C}=\Big\{u \in W^{1,\mathcal{H}}(\Omega)\colon u^{\pm}\neq 0,\, \left\langle \mathcal{E}'(u),u^+ \right\rangle= \left\langle \mathcal{E}'(u),-u^- \right\rangle=0 \Big\},\] whereby $\mathcal{C}$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.