On global bifurcation for the nonlinear Steklov problems

TL;DR

This paper investigates the bifurcation phenomena of nonlinear Steklov problems involving the p-Laplace operator, establishing the existence of solution continua emanating from the first eigenvalue under certain conditions.

Contribution

It extends bifurcation theory to nonlinear Steklov problems with indefinite weights and general growth conditions, providing new existence results for solution continua.

Findings

Existence of a bifurcation continuum from the first eigenvalue.

Application of Lorentz-Zygmund space conditions for weights.

General growth conditions at zero and infinity for the nonlinearity.

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*}