We study boson stars in a theory of complex scalar field coupled to Einstein gravity with the potential: V (|Φ|) := m 2 |Φ| 2 + 2λ |Φ| (where m 2 and λ are positive constant parameters). This could be considered either as a theory of massive complex scalar field coupled to gravity in a conical potential or as a theory in the presence of a potential which is an overlap of a parabolic and a conical potential. We study our theory with positive as well as negative values of the cosmological constant Λ . Boson stars are found to come in two types, having either ball-like or shell-like charge density. We have studied the properties of these solutions and have also determined their domains of existence for some specific values of the parameters of the theory. Similar solutions have also been obtained by Hartmann, Kleihaus, Kunz, and Schaffer, in a V-shaped scalar potential.Keywords Gravity Theories · Boson stars · Boson shells · Q-balls · Q-shells A study of boson shells and boson stars in scalar electrodynamics with a self-interacting complex scalar field Φ coupled to Einstein gravity is of a very wide interest in the gravity theories[1]- [26] . Hartmann, Kleihaus, Kunz, and Schaffer (HKKS) [2,3] have recently studied boson stars in a theory of complex scalar field coupled to Einstein gravity in a V-shaped scalar potential: V (ΦΦ * ) ≡ V (|Φ|) = λ c |Φ| (where λ c is a constant). They have found that the boson stars come in two types, having either ball-like or shell-like charge density. They have studied the properties of these solutions and have also determined their domains of existence.Actually, the boson stars represent localized self-gravitating solutions [6,7,8], that have been considered in many different contexts [9,10,11,12,13]. To obtain boson stars, typically a complex scalar field Φ is considered. The U(1) invariance of the theory then provides a conserved Noether current.The properties of the boson stars depend strongly on the self-interaction employed. In particular, as discussed by Lee and collaborators [14] and by Coleman [15], the existence of a flat space-time limit of these localized solutions, i.e., the existence of Q-balls, puts constraints on the types of self-interaction possible.Arodz and Liz showed, that besides Q-balls another type of localized solution could also arise in flat space, when a V-shaped self-interaction is employed in the presence of a gauge field [16,17,18]. In this type of solution, the energy density is no longer ball-like as in the case of Q-balls, but it is instead shell-like.