We solve the equation of the equilibrium of the gravitating body, with a
polytropic equation of state of the matter P=Kργ, with
γ=1+1/n, in the frame of the Newtonian gravity, with non-zero
cosmological constant Λ. We consider the cases with n=1,1.5,3
and construct series of solutions with a fixed value of Λ. For each
value of n, the non-dimensional equation of the static equilibrium has a
family of solutions, instead of the unique solution of the Lane-Emden equation
at Λ=0. The equilibrium state exists only for central densities
ρ0 larger than the critical value ρc. There are no static solutions
at ρ0<ρc. We find the values of ρc for each value of n and
show that the presence of dark energy decrease the dynamic stability of the
configuration. We apply our results for analyzing the possibility of existence
of equilibrium states for cluster of galaxies in the present universe with
non-zero Λ.Comment: submitted to Astron. A