Global bifurcation for N-dimensional p-Laplacian problem and its applications. Issue 12 (2nd December 2022) - Ye, Fumei; Han, Xiaoling
Abstract : In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem 1 − 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 2 − 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.