On global bifurcation for the nonlinear Steklov problems - Anoop, Thazhe Veetil; Biswas, Nirjan (Topological Methods in Nonlinear Analysis, 2021)
For p (1, ∞), for an integer N ≥ 2 and for a bounded Lipschitz domain Ω ^N, we consider the following nonlinear Steklov bifurcation problem -Δ_p φ = 0 in Ω, | abla φ|^p-2 φu = λ ( g φ^p-2φ + f r(φ) ) on Ω, where Δ_p is the p-Laplace operator, g,f L^1(Ω) are indefinite weight functions and r C(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 in Ω, | abla φ|^p-2 φu = λ g φ^p-2φ on Ω.