ON GLOBAL BIFURCATION FOR THE NONLINEAR STEKLOV PROBLEMS

For p ∈ (1, ∞), for an integer N ≥ 2 and for a bounded Lip-schitz domain Ω ⊂ RN, we consider the following nonlinear Steklov bifurcation problem where ∆p is the p-Laplace operator, g, f ∈ L1 (∂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 (λ1, 0), where λ1 is the first eigenvalue of the following nonlinear Steklov eigenvalue problem ∂φ.