Polytropic configurations with non-zero cosmological constant

We solve the equation of the equilibrium of the gravitating body, with a polytropic equation of state of the matter \(P=K\rho^{\gamma}\), with \(\gamma=1+1/n\), in the frame of the Newtonian gravity, with non-zero cosmological constant \(\Lambda\). We consider the cases with \(n=1,\,\,1.5,\,\,3\) and construct series of solutions with a fixed value of \(\Lambda\). 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 \(\Lambda=0\). The equilibrium state exists only for central densities \(\rho_0\) larger than the critical value \(\rho_c\). There are no static solutions at \(\rho_0 < \rho_c\). We find the values of \(\rho_c\) 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 \(\Lambda\).