The authors study the nonlinear elliptic boundary value problem \[ \Delta u+\sigma\cdot\mu\cdot\exp u=0,\quad u\mid_{\partial\Omega}=0, \] where \(\mu=\left(\int_ \Omega\exp u\cdot dV\right)^{-1}\) and \(\sigma\) is a positive constant. This problem describes the gravitational potential of self-gravitating thermodynamically equilibrium gas filling up a bounded domain \(\Omega\subset\mathbb{R}^ 3\). Using the Schauder techniques, it is shown, that the problem has a solution for all sufficiently small \(\sigma\) and has no solution, if \(\sigma>3\left(\int_{\partial\Omega}{dS\over(x,n)}\right)^{-1}\), where \(n\) denotes the exterior unit normal to \(\partial\Omega\).