We consider the following weighted eigenvalue problem in the exterior domain: (Formula presented.),where Δp is the p-Laplace operator with p>1, and (Formula presented.) is the exterior of the closed unit ball in RN with N≥1. There is no restriction on the dimension N in terms of $$p, i.e., we allow both 1<p<N and p≥N. The weight function K is locally integrable on (Formula presented.) and is allowed to change its sign. For some appropriate choice of $$w$$w, a positive weight function on the interval (1,∞), we prove that the Beppo-Levi space (Formula presented.) is compactly embedded into the weighted Lebesgue space Lp(Formula presented.). The existence of the positive eigenvalue for the above problem is proved for K such that (Formula presented.) 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. © 2014, Springer-Verlag Berlin Heidelberg.