Two global several-parameter bifurcation theorems and their applications

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equationu=L(λ)u+H(λ,u),(λ,u)∈Rm×X, where m is a positive integer, X   is a Banach space, L(⋅):X→XL(⋅):X→X is a (positively) homogeneous compact operator and H:Rm×X→XH:Rm×X→X is compact with H=o(‖u‖)H=o(‖u‖) near u=0u=0 uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.