  We derive and analyze the coupled equations of quadratic approximation of the
Bogoliubov model for a weakly interacting Bose gas. The first equation
determines the condensate density as a variational parameter and ensures the
minimum of the grand thermodynamic potential. The second one provides a
relation between the total number of particles and chemical potential. Their
consistent theoretical analysis is performed for a number of model interaction
potentials including contact (local) and nonlocal interactions, where the
latter provide nontrivial dependencies in momentum space. We demonstrate that
the derived equations have no solutions for the local potential, although they
formally reproduce the well-known results of the Bogoliubov approach. At the
same time, it is shown that these equations have the solutions for physically
relevant nonlocal potentials. We show that in the regimes close to experimental
realizations with ultracold atoms, the contribution of the terms originating
from the quadratic part of the truncated Hamiltonian to the chemical potential
can be of the same order of magnitude as from its $c$-number part. Due to this
fact, in particular, the spectrum of single-particle excitations in the
quadratic approximation acquires a gap. The issue of the gap is also discussed.
