On global bifurcation for the nonlinear Steklov problems

Abstract

For $p\in (1,\infty ),$ for an integer $N\ge 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λ\left( g |ϕ|^{p-2}ϕ+ f r(ϕ) \right) \; \text{on} \ \partial Ω, \end{aligned} \end{equation*} where ${\Delta }_p$ is the $p$-Laplace operator, $g,f\in L^1(\partial \Omega )$ are indefinite weight functions and $r\in C(\mathbb{R})$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f,g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $({\lambda }_1,0)$, where ${\lambda }_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λg |ϕ|^{p-2}ϕ\ \text{on} \ \partial Ω. \end{aligned} \end{equation*}