Global Bifurcation Structure For The Solution Of Weighted P-harmonic Equation

In this paper, we discuss the global bifurcation result for the p — harmonic operatorin the weighted Sobolev space with Navier boundary conditionDenoteFor any , we definewhere w(x) = {wi(x)}_iⁿ=0 are vector-valued functions, and W₀^1,p(Ω,w) denotes the weighted Sobolev space (the definition will be given in the section 2).Let Ω∈Rⁿ be a bounded domain with smooth boundary σΩ. For p G (1, ∞), we consider the nonlinear eigenvalue problemWe prove that (1.1) has a principle positive eigenvalue λ₁ = λ₁(p) which is simple and isolated. Moreover, we prove that there exists strictly positive eigenfunctionu₁ = u₁(p) in Ω associated with λ₁(p) which satisfies σu₁/σn< 0 on σΩ We also showthat p → λ₁(p) is a continuous function in (1,∞).Next we consider the following boundary value problemWhere the function g(x, A, u) denotes the high order term of (1.2), and satisfies some proper growth condition. By means of the Leray — Schauder degree theory, we prove that (Ai(p),0) is a bifurcation point of (1.2), and the global bifurcation result for (1.2) is achieved by the standard global bifurcation theory in [10].