Singular Solutions of the Tolman Oppenheimer Volkoff Equation with a Cosmological Constant Classification and Properties

We study the Tolman-Oppenheimer-Volkoff equation in the presence of a cosmological constant for general thermodynamically consistent equations of state, without imposing regularity at the center. Formulating the problem as an initial value system integrated from an outer boundary inwards, we obtain a general classification of solutions and show that singular configurations dominate the solution space. We demonstrate that all singular solutions share a universal geometric structure and give rise to spacetimes that are bounded-acceleration complete, indicating that the associated singularities are comparatively mild. Our results extend the classification previously obtained for Λ=0 and reveal qualitatively new features for $\mathrm{\Lambda }\mathrm{\ne }0$. For $\mathrm{\Lambda }<0$, we identify solutions with approximate horizon structures that mimic black holes in equilibrium with their Hawking radiation. For $\mathrm{\Lambda }>0$, we find four distinct classes of solutions with cosmological horizons, distinguished by the behavior of their temperature gradients.