We study boundary value problems of the form {-δu=λK({pipe}x{pipe})f(u),x∈Ωu=0if{pipe}x{pipe}=r0u→0as{pipe}x{pipe}→∞, where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n>2,{pipe}x{pipe}>r0}, K belongs to a class of C1 functions such that limr→∞K(r)=0, and f belongs to a class of C1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.}, author_keywords={Existence; Exterior domains; Falling zeros; Semipositone problems; Uniqueness}, references={Ali, I., Castro, A., Shivaji, R., Uniqueness and stability of positive solutions for semipositone problems in a ball (1993) Proc. Amer. Math. Soc., 117, pp. 775-782; Ambrosetti, A., Arcoya, D., Biffoni, B., Positive solutions for some semipositone problems via bifurcation theory (1994) Differential Integral Equations, 7 (3), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone boundary value problems (1996) Proc. Amer. Math. Soc., 124 (3), pp. 757-763; Castro, A., Gadam, S., Shivaji, R., Positive solution curves of semipositone problems with concave nonlinearities (1997) Proc. Roy. Soc. Edinburgh Sect. A, 127 (5), pp. 921-934; Castro, A., Garner, J.B., Shivaji, R., Existence results for classes of sublinear semipositone problems (1993) Results Math., 23, pp. 214-220; Castro, A., Hassanpour, M., Shivaji, R., Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity (1995) Comm. Partial Differential Equations, 20 (11-12), pp. 1927-1936; Castro, A., Sankar, L., Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains (2012) J. Math. Anal. Appl., 394 (1), pp. 432-437; Castro, A., Shivaji, R., Nonnegative solutions for a class of nonpositone problems (1998) Proc. Roy. Soc. Edinburgh Sect. A, 108, pp. 291-302; Castro, A., Shivaji, R., Nonnegative solutions for a class of radially symmetric nonpositone problems (1989) Proc. Amer. Math. Soc., 106 (3), pp. 735-740; Castro, A., Shivaji, R., Positive solutions for a concave semipositone Dirichlet problem (1998) Nonlinear Anal., 31 (1-2), pp. 91-98; Dancer, E.N., Shi, J., Uniqueness and nonexistence of positive solutions to semipositone problems (2006) Bull. London Math. Soc., 38 (6), pp. 1033-1044; Habets, P., Zanolin, F., Upper an lower solutions for a generalized Emden-Fowler equation (1994) J. Math. Anal. Appl., 181, pp. 684-700; Lee, E., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24, pp. 861-875; Philippe Clement, I., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Serrin, J., (1971), Nonlinear equations of second order, in: A.M.S. Sympos. Partial Differential Equations, Berkeley, August}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , NC 27412, United States; email: shivaji@uncg.edu}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},