On global bifurcation for the nonlinear Steklov problems - Anoop, Thazhe Veetil; Biswas, Nirjan (Topological Methods in Nonlinear Analysis, 2021)
For $p \in (1, ∞)$, for an integer $N ≥ 2$ and for a bounded Lipschitz domain $Ω \subset \R^N$, we consider the following nonlinear Steklov bifurcation problem$$-Δ_p φ = 0 \quad \text{in } Ω, \qquad | abla φ|^{p-2} \frac{\partial φ}{\partial u} = λ \big( g \abs{φ}^{p-2}φ + f r(φ) \big) \quad \text{on } \partial Ω,$$where $Δ_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial Ω)$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 $(λ_1,0)$,where $λ_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem$$-Δ_p φ = 0 \quad \text{in } Ω, \qquad | abla φ|^{p-2} \frac{\partial φ}{\partial u} = λ g \abs{φ}^{p-2}φ \quad \text{on } \partial Ω.$$