Few-Body Syst (2018) 59:6 https://doi.org/10.1007/s00601-018-1328-4 Sanjeev Kumar · Usha Kulshreshtha · Daya Shankar Kulshreshtha Boson Stars and Boson Shells Received: 6 December 2017 / Accepted: 30 December 2017 © Springer-Verlag GmbH Austria, part of Springer Nature 2018 Abstract In this work we present a broad formalism for a study of the models of black holes, boson stars, boson shells and wormholes. The studies of boson stars and boson shells in a theory involving Scalar field, U (1) gauge field and a shelf interacting scalar potential coupled to gravity in the presence of a cosmological constant Λ are presented in details. In this work we present a rather general albeit broad formalism for a study of a wide class of gravity models [1–9] in general relativity for describing a class of the models of black holes, boson stars, boson shells and even the wormholes [1–9]. For such a study one needs to consider a specific action, the metric and the appropriate boundary conditions defining the theory of interest. In the following we present some specific studies of boson stars and boson shells within the above framework in the presence of a cosmological constant Λ which could take positive as well as negative values corresponding respectively to the de Sitter (dS) and Anti de Sitter (AdS) spaces. The theories with positive values of Λ are relevant from the observational point of view as they describe a more realistic description of the compact stars in the Universe since all the observations seem to indicate the existence of a positive cosmological constant and the theories with negative values of Λ are meaningful in the context of AdS-CFT correspondence. The action of a theory describing the boson stars and boson shells in a theory of complex scalar field coupled to the U (1) gauge field A μ and the gravity in the presence of a self interacting scalar potential and cosmological constant Λ reads [6–8]: S =  R − 2Λ 16π G − 1 4 F μν F μν − ( D μ Φ ) ∗ ( D μ Φ ) − V (|Φ|)  √ −gd 4 x , D μ Φ = (∂ μ Φ + ieA μ Φ), F μν = (∂ μ A ν − ∂ ν A μ ), V (|Φ|) = m 2 |Φ| 2 + λ|Φ|. (1) Here R is Ricci curvature scalar, G is Newton’s Gravitational constant, Λ is cosmological constant and m and λ are constant parameters. Also, g = det(g μν ) where g μν is the metric tensor and the asterisk in the above equation denotes complex conjugation. Using the variational principle, equations of motion are obtained as: Talk presented by D.S. Kulshreshtha. S. Kumar (B) · D. S. Kulshreshtha Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India E-mail: sanjeev.kumar.ka@gmail.com D. S. Kushreshtha E-mail: dskulsh@gmail.com U. Kulshreshtha Department of Physics, Kirori Mal College, University of Delhi, Delhi 110007, India E-mail: ushakulsh@gmail.com