Main Properties of the Solution

In this section we will state and prove a few important properties of the solution. We will assume that, given certain values of $r_{1}$, $r_{2}$ and $r_{M}$, the corresponding solution exists. In other words, we are assuming that a solution of Equation (57) for $\Upsilon_{0}$ can be found, thus determining $\rho_{0}$, which includes establishing the strict positivity of the cubic polynomial within the square roots in the denominators, and that a corresponding function $z(r)$ is therefore determined via Equation (52). This then implies that the solutions for both $\lambda(r)$ and $\nu(r)$, as well as for $P(r)$, are all determined, with all the boundary conditions duly satisfied. A simpler way to put this is to say that we are establishing the most important properties of all existing solutions of the problem. For easy reference, we state the complete solution explicitly in Table 1, where we have that $\rho_{0}$ is determined algebraically via Equation (57), $z(r)$ is determined by Equation (52), and $r_{\mu}$ is given by Equation (31). We will start by the discussion of the presence of the singularity at the origin.

Table 1: Summary of the results.