Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators

In this paper, we analyze an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to L-infinity, which implies C-1,C-alpha smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray-Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.