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The main result characterizes the global phenomena of components with one-sign solutions for the Kirchhoff type problem { − M (∫ Ω | ∇ u | 2 d x) Δ u = λ a (x) u (x) + f (x , u , λ) + g (x , u , λ) in Ω , u = 0 on ∂ Ω involving sign-changing weight function, where Ω is a smooth bounded domain in R N , λ ≠ 0 is a real parameter, a ∈ L ∞ (Ω) with a ≢ 0 , f , g ∈ C (Ω ‾ × R 2 , R). The method relies upon the bifurcation theory. According to the behaviors of f at 0, we determine the range of parameter λ for the one-sign solutions of the above problem with indefinite weight function a. Moreover, the global bifurcation phenomenon of this problem can be obtained mainly depending on f satisfying the signum condition s f (x , s , λ) < 0 for s ≠ 0. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem − d i v (φ p (∇ y)) = λ a (x) φ p (y) + a (x) f (x , y , λ) + g (x , y , λ) , i n B , y = 0 , o n ∂ B , where λ is a parameter, f , g ∈ C (B × R × R , R). Let B be a unit open ball of R N with a smooth boundary ∂ B. We shall show that there are two distinct unbounded continua C k + and C k − , consisting of the bifurcation branch C k if f is not necessarily differentiable at the origin with respect to φ p (y) , and there are two distinct unbounded continua D k + and D k − , consisting of the bifurcation branch D k if f is not necessarily differentiable at infinity with respect to φ p (y). As the applications of the above result, we shall prove more details about the existence and multiplicity results of sign-changing solutions for the elliptic problem − d i v (φ p (∇ y)) = λ a (x) f (y) + a (x) g (y) , i n B , y = 0 , o n ∂ B , where f , g ∈ C (R , R) and g is not necessarily differentiable at the origin and infinity with respect to φ p (y). Furthermore, by using a comparison theorem, we also obtain a non-existence result of nodal solutions to the above problem. [ABSTRACT FROM AUTHOR]