  In this work we study the boson stars and boson shells in a theory involving
massive complex scalar fields coupled to the U(1) gauge field and gravity in a
conical potential in the presence of a cosmological constant ${\Lambda}$ which
we treat as a free parameter taking positive and negative values and thereby
allowing us to study the theory in the de Sitter and Anti de Sitter spaces
respectively. 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 Kleihaus, Kunz, Laemmerzahl and List in a theory involving massless
complex scalar fields coupled to the U(1) gauge field and gravity in a conical
potential in the absence of a cosmological constant ${\Lambda}$.
