We investigate the following eigenvalue problem {−div(L(x)|∇u|p−2∇u)=λK(x)|u|p−2uin AR1 R2,u=0on ∂AR1 R2, where AR1 R2:={x∈RN:R1<|x|<R2} (0<R1<R2≤∞), λ>0 is a parameter, the weights L and K are measurable with L positive a.e. in AR1 R2 and K possibly sign-changing in AR1 R2. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for u(x) and ∇u(x) as |x|→R1 + or R2 − are also investigated. © 2018 Elsevier Inc.}, author_keywords={Asymptotic behavior; Exterior domain; Maximum principles; Regularity; The first eigenvalue; The weighted p-Laplacian; Variational method}, funding_details={Ministerstvo Školství, Mládeže a TělovýchovyMinisterstvo Školství, Mládeže a Tělovýchovy, MŠMT, LO1506}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 18-032523S}, funding_text_1={P. Drábek and A. Sarkar were partly supported by the Grant Agency of the Czech Republic , project no. 18-032523S . K. Ho and A. Sarkar were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports .}, funding_text_2={P. Dr?bek and A. Sarkar were partly supported by the Grant Agency of the Czech Republic, project no. 18-032523S. K. Ho and A. Sarkar were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.}, references={Agudelo, O., Drábek, P., Anisotropic semipositone quasilinear problems (2017) J. Math. Anal. Appl., 452 (2), pp. 1145-1167; Anoop, T.V., Drábek, P., Sankar, L., Sasi, S., Antimaximum principle in exterior domains (2016) Nonlinear Anal., 130, pp. 241-254; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. 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