\nonumber\\
&&\qquad\qquad\quad\! \;+\,\frac{\,2r^{2}}{3}\!\int_{r}^{\,\infty} \frac{\,\xi-3m}{r^3} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}dr\,.
\label{Byparts}\end{aligned}$$ Taking into account (\[cond1\]) and (\[ineq1\]), we have $$\label{ineq1}
A(r)\,>\,\frac{\xi-3m}{r}\mathrm{e}^{\Phi} \quad\text{for all}\;\; r>0\,.
\nonumber$$

As it can be seen from (\[A’\]), this inequality implies that $A'>0$ in the region where $\xi-3m>0$.

For a SFNS, the existence of a minimum follows directly from (\[3m&lt;xi0\]), (\[asymp1\]), and (\[asymp2\]). Suppose that there are other extrema. Let $r_0$ be the local minimum point farthest from the origin $r=0$, and $r_m$ be the local maximum point nearest to $r_0$, that is, there are no local extrema in the interval $(r_m,r_0)$. Then $A'(r_0)=A'(r_m)=0$, $A''(r_0)>0$, and $A''(r_m)<0$, so that (\[A"\]) gives $$\label{cond2}
\left(\frac{\xi-3m}{r}\, \mathrm{e}^{\Phi}\right)'_{r=r_0}< 0\,,\;\;\; \left(\frac{\xi-3m}{r}\, \mathrm{e}^{\Phi}\right)'_{r=r_m}> 0\,.
\nonumber$$ Consequently, the minimum at $r=r_0$ is unique if $$\label{cond3}
\left(\frac{\xi-3m}{r}\, \mathrm{e}^{\Phi}\right)'\leq 0\,,\quad 0<r<r_0\,.$$ It is important to note that this condition is quite natural and means simply that the configurations of a scalar field is sufficiently compact and the function $\mathrm{e}^{\Phi}$ varies sufficiently smoothly (has no steep jumps). A more detailed characterization in terms of the field function $\phi$ turns out to be still more complicated, but numerous example show that the condition (\[cond3\]) holds for physically reasonable configurations. Thus we have stated the following proposition.

\[proposition2\] In a SFNS spacetime defined 