Global bifurcation, Berestycki’s conjecture and one-sign solutions for pp-Laplacian

In this paper, we shall establish the unilateral global bifurcation result from an interval for a class of high-dimensional pp-Laplacian problems. As an application of the above result, we shall prove the existence of the principal half-eigenvalues for a class of half-quasilinear eigenvalue problems. Our result partially answers the open problem proposed by Berestycki. Moreover, we shall investigate the existence of one-sign solutions for a class of high-dimensional pp-Laplacian problems with non-p−p−1-homogeneous nonlinearities.