Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

Abstract

In this paper we consider degenerate Kirchhoff-type equations of the form$-\varphi \left(\Xi \right(u\left)\right)\left(A\right(u)-|u{|}^{p-2}u)=f(x,u)\text{in }\Omega ,\varphi \left(\Xi \right(u\left)\right)B\left(u\right)\cdot \nu =g(x,u)\text{on }\partial \Omega ,$ where $\Omega \subseteq R^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary ∂Ω, $A$ denotes the double phase operator given by$A\left(u\right)=\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u+\mu \left(x\right)\left|\nabla u{|}^{q-2}\nabla u\right)$ for $u\in W^{1,H}\left(\Omega \right)$, $\nu \left(x\right)$ is the outer unit normal of Ω at $x\in \partial \Omega$,$B\left(u\right)=|\nabla u{|}^{p-2}\nabla u+\mu (x\left)\right|\nabla u{|}^{q-2}\nabla u,\Xi \left(u\right)=\underset{\Omega }{\int }(\frac{|\nabla u{|}^p+|u{|}^p}{p}+\mu (x\left)\frac{|\nabla u{|}^q}{q}\right)dx,$ $1<p<N$, $p<q<p^{⁎}=\frac{Np}{N-p}$, $0\le \mu (\cdot )\in L^{\infty }\left(\Omega \right)$, $\varphi \left(s\right)=a+b{s}^{\zeta -1}$ for $s\in R$ with $a\ge 0$, $b>0$ and $\zeta \ge 1$, and $f:\Omega \times R\to R$, $g:\partial \Omega \times R\to R$ 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 $E:W^{1,H}\left(\Omega \right)\to R$ over the constraint set$\begin{array}{cc}& C=\{u\in W^{1,H}(\Omega ):u^{\pm }\ne 0,\\ & 〈E^{\prime }\left(u\right),u^{+}〉=〈E^{\prime }\left(u\right),-u^{-}〉=0\},\end{array}$ whereby $C$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.