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 $\Omega \subset \mathbb R^N$, we consider the following nonlinear Steklov bifurcation problem$$-\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda \big( g |\phi|^{p-2}\phi + f r(\phi) \big) \quad \text{on } \partial \Omega,$$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$$-\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda g |\phi|^{p-2}\phi \quad \text{on } \partial \Omega.$$