We consider the problem {−Δu=λK(x)f(u)in B1 c,u=0on ∂B1,u(x)→0as |x|→∞, where B1 c={x∈Rn||x|>1},n>2, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter λ. Existence results to similar problems in Rn are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions. © 2020}, author_keywords={Asymptotically linear reaction term; Infinite semipositone; Method of sub and super solutions; Schauder fixed point theorem; Semilinear elliptic problems; Unbounded domains}, funding_details={Department of Science and Technology, Ministry of Science and Technology, IndiaDepartment of Science and Technology, Ministry of Science and Technology, India, DST, DST/INSPIRE/04/2014/002833}, funding_text_1={This research was supported by the INSPIRE faculty award, awarded by the Department of Science and Technology , Government of India ( DST/INSPIRE/04/2014/002833 ).}, references={Abebe, A., Chhetri, M., Sankar, L., Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains (2014) Bound. Value Probl., 2014. , 9 pp; Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces (1976) SIAM Rev., 18 (4), pp. 620-709; Ávila, A.I., Brock, F., Asymptotics at infinity of solutions for p-Laplace equations in exterior domains (2008) Nonlinear Anal., 69 (5-6), pp. 1615-1628; 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; Chhetri, M., Drábek, P., Principal eigenvalue of p-Laplacian operator in exterior domain (2014) Results Math., 66 (3-4), pp. 461-468; Cui, S., Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems (2000) Nonlinear Anal., 41 (1-2), pp. 149-176; Drábek, P., Sankar, L., Singular quasilinear elliptic problems on unbounded domains (2014) Nonlinear Anal., 109, pp. 148-155; Fleckinger-Pellé, J., Gossez, J.-P., de Thélin, F., Principal eigenvalue in an unbounded domain and a weighted Poincaré inequality (2006) Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, pp. 283-296. , Birkhäuser Basel; Hai, D.D., On an asymptotically linear singular boundary value problems (2012) Topol. Methods Nonlinear Anal., 39 (1), pp. 83-92; Hai, D.D., Sankar, L., Shivaji, R., Infinite semipositone problems with asymptotically linear growth forcing terms (2012) Differ. Integral Equ., 25 (11-12), pp. 1175-1188; Hai, D.D., Williams, J.L., Positive radial solutions for a class of quasilinear boundary value problems in a ball (2012) Nonlinear Anal., 75 (4), pp. 1744-1750; Ko, E., Lee, E.K., Shivaji, R., Multiplicity results for classes of singular problems on an exterior domain (2013) Discrete Contin. Dyn. Syst., 33 (11-12), pp. 5153-5166; Lee, E.K., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differ. Integral Equ., 24 (9-10), pp. 861-875; Rajendran, D., Tyagi, J., Existence of nonnegative solutions to positone-type problems in R̂N with indefinite weights (2010) Electron. J. Differ. Equ., 2010. , 16 pp}, correspondence_address1={Sankar, L.; Indian Institute of TechnologyIndia; email: lakshmi@iitpkd.ac.in}, publisher={Academic Press Inc.}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},