We study the nature of phase transitions in a self-gravitating classical gas in the presence of a central body. The central body can mimic a black hole at the center of a galaxy or a rocky core (protoplanet) in the context of planetary formation. In the chemotaxis of bacterial populations, sharing formal analogies with self-gravitating systems, the central body can be a supply of "food" that attracts the bacteria (chemoattractant). We consider both microcanonical (fixed energy) and canonical (fixed temperature) descriptions and study the inequivalence of statistical ensembles. At high energies (respectively, high temperatures), the system is in a "gaseous" phase and at low energies (respectively, low temperatures) it is in a condensed phase with a "cusp-halo" structure, where the cusp corresponds to the rapid increase of the density of the gas at the contact with the central body. For a fixed density ρ_{*} of the central body, we show the existence of two critical points in the phase diagram, one in each ensemble, depending on the core radius R_{*}: for small radii R_{*}<R_{*}^{MCP}, there exist both microcanonical and canonical phase transitions (that are zeroth and first order); for intermediate radii R_{*}^{MCP}<R_{*}<R_{*}^{CCP}, only canonical phase transitions are present; and for large radii R_{*}>R_{*}^{CCP}, there is no phase transition at all. We study how the nature of these phase transitions changes as a function of the dimension of space. We also discuss the analogies and the differences with phase transitions in the self-gravitating Fermi gas [P. H. Chavanis, Phys. Rev. E 65, 056123 (2002)1063-651X10.1103/PhysRevE.65.056123].