We consider the nonlinear eigenvalue problem where Δp is the p-Laplacian operator, Ω is a connected domain in RN with N > p and the weight function g is locally integrable. We obtain the existence of a unique positive principal eigenvalue for g such that g+ lies in certain subspace of weak-LN/p(Ω). The radial symmetry of the first eigenfunctions are obtained for radial g, when Ω is a ball centered at the origin or RN. The existence of an infinite set of eigenvalues is proved using the Ljusternik-Schnirelmann theory on C1 manifolds. © 2011 Texas State University - San Marcos.