  We define a formalism of a self-consistent description of the ground state of
a weakly interacting Bose system, accounting for higher order terms in
expansion of energy in the diluteness parameter. The approach is designed to be
applied to a Bose-Bose mixture in a regime of weak collapse where quantum
fluctuations lead to stabilization of the system and formation of quantum
liquid droplets. The approach is based on the Generalized Gross -- Pitaevskii
equation accounting for quantum depletion and anomalous density terms. The
equation is self-consistently coupled to modified Bogoliubov equations.
  The modification we introduce resolves the longstanding issue of missing
phonon-branch excitations when higher order terms are included. Our method
ensures a gapless phononic low-energy excitation spectrum, crucial to correctly
account for quantum fluctuations. We pay particular attention to the case of
droplets harmonically confined in some directions. The method allows to
determine the Lee-Huang-Yang-type contribution to the chemical potential of
inhomogeneous droplets when the local density approximation fails.
