On global bifurcation for the nonlinear Steklov problems

Abstract

For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\ |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} &= \lambda \left( g |\phi|^{p-2}\phi + f r(\phi) \right) \; \text{on} \ \partial \Omega, \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} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\ |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} &= \lambda g |\phi|^{p-2}\phi \ \text{on} \ \partial \Omega. \end{aligned} \end{equation*}