We consider the following weighted eigenvalue problem in the exterior domain:

-Apu = K(x)|u|P-2 u in B,

โ u = 0 on 3B1,

where A, is the p-Laplace operator with p > 1, and B is the exterior of the closed unit ball in RN with N > 1. There is no restriction on the dimension N in terms of p, i.e., we allow both 1<p< N and p > N. The weight function K is locally integrable on B and is allowed to change its sign. For some appropriate choice of w, a positive weight function on the interval (1,∞), we prove that the Beppo-Levi space D (B) is compactly embedded into the weighted Lebesgue space LP (B; w(x)). The existence of the positive eigenvalue for the above problem is proved for K such that suppK is of non-zero measure and K <w. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions.