We consider the antimaximum principle for the p-Laplacian in the exterior domain: {-Δpu=λK(x)|u|p-2u+h(x)in B1c,u=0on ∂B1, where Δp is the p-Laplace operator with p>1,λ is the spectral parameter and B1c is the exterior of the closed unit ball in RN with N ≥ 1. The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N=1 and p=2, we provide a necessary and sufficient condition on h for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems. © 2015 Elsevier Ltd.}, author_keywords={Exterior domains; Local and global antimaximum; p-Laplacian; Positive eigenfunctions; principle; Regularity results}, keywords={Boundary value problems; Eigenvalues and eigenfunctions; Laplace transforms, Exterior domain; Local and global antimaximum; P-Laplacian; principle; Regularity results, Nonlinear equations}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The second author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S.}, funding_text_2={The first, third and fourth authors work is funded by the project EXLIZ-CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.}, references={Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. Partial Differential Equations, 53 (3-4), pp. 961-975; Birindelli, I., Hopf's lemma and anti-maximum principle in general domains (1995) J. Differential Equations, 119 (2), pp. 450-472; Clément, P., Peletier, L.A., An anti-maximum principle for second-order elliptic operators (1979) J. Differential Equations, 34 (2), pp. 218-229; Drábek, P., Huang, Y.X., Bifurcation problems for the p-Laplacian in RN (1997) Trans. Amer. Math. Soc., 349 (1), pp. 171-188; Fleckinger-Pellé, J., Gossez, J.-P., De Thélin, F., Antimaximum principle in RN: Local versus global (2004) J. Differential Equations, 196 (1), pp. 119-133; Fučík, S., Nečas, J., Souček, J., Souček, V., (1973) Spectral Analysis of Nonlinear Operators, 346. , Lecture Notes in Mathematics Springer-Verlag Berlin, New York; Godoy, T., Gossez, J.-P., Paczka, S., On the antimaximum principle for the p-Laplacian with indefinite weight (2002) Nonlinear Anal., 51 (3), pp. 449-467; Hess, P., An anti-maximum principle for linear elliptic equations with an indefinite weight function (1981) J. Differential Equations, 41 (3), pp. 369-374; Hörmander, L., Lions, J.L., Sur la complétion par rapport à une intégrale de Dirichlet (1956) Math. Scand., 4, pp. 259-270; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Shi, J., A new proof of anti-maximum principle via a bifurcation approach (2005) Results Math., 48 (1-2), pp. 162-167; Stavrakakis, N.M., De Thélin, F., Principal eigenvalues and anti-maximum principle for some quasilinear elliptic equations on RN (2000) Math. Nachr., 212, pp. 155-171; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202; Ziemer, W.P., Sobolev spaces and functions of bounded variation (1989) Weakly Differentiable Functions, 120. , Graduate Texts in Mathematics Springer-Verlag New York}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Univerzitní 22, Czech Republic}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},