Weighted quasilinear eigenvalue problems in exterior domains

We consider the following weighted eigenvalue problem in the exterior domain: - Δ p u = λ K ( x ) | u | p - 2 u in B 1 c , u = 0 on ∂ B 1 , where Δ p is the p -Laplace operator with p > 1 , and B 1 c is the exterior of the closed unit ball in R N 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 1 c 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 0 1 , p ( B 1 c ) is compactly embedded into the weighted Lebesgue space L p ( B 1 c ; w ( | x | ) ) . The existence of the positive eigenvalue for the above problem is proved for K such that supp K + 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.