  A self-consistent model of the superfluid (SF) state of a Bose liquid with
strong interaction between bosons is considered, in which at T=0, along with a
weak single-particle Bose-Einstein condensate (BEC), there exists an intensive
pair coherent condensate (PCC) of bosons, analogous to the Cooper pair
condensate of fermions. Such a PCC emerges due to an effective attraction
between bosons in some regions of momentum space, which results from an
oscillating sign-changing momentum dependence of the Fourier component V(p) of
the interaction potential. The collective many-body effects of renormalization
("screening") of the initial interaction, which are described by the bosonic
polarization operator \Pi(p,\omega), lead to a suppression of the repulsion
[V(p)>0] and an enhancement of the effective attraction [V(p)<0] in the
respective domains of nonzero momentum transfer. The ratio of the BEC density
n_0 to the total particle density n of the Bose liquid is used as a small
parameter of the model. A closed system of nonlinear integral equations for the
normal and anomalous self-energy parts is obtained with account for the terms
of first order in the BEC density. In the framework of the ``soft spheres''
model with the single fitting parameter--the value of the repulsion potential
at r=0, a theoretical quasiparticle spectrum E(p) is obtained, which is in good
accordance with the experimental spectrum E_{exp}(p) of elementary excitations
in superfluid $^4$He.
