  Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial
x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*}
be a second order elliptic operator and consider the reaction-diffusion
equation with Neumann boundary condition, \begin{equation*} \begin{aligned}
&Lu=\Lambda u^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\
\text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned}
\end{equation*} where $p\in(0,1)$, $d\ge2$, $h$ and $\Lambda$ are continuous
positive functions, $D\subset R^d$ is bounded, and $\bar n$ is the unit inward
normal to the domain $\mathbb{R}^d-\bar D$. Consider also the same equations
with the Neumann boundary condition replaced by the Dirichlet boundary
condition; namely, $u=h$ on $\partial D$. The solutions to the above equations
may possess a free boundary. When $D=\{|x|<R\}$ and $L$ and $\Lambda$ are
radially symmetric, we write the solution as $u(r)$ with $r=|x|$ and define the
radius of the free boundary by $r^*(h)=\inf\{r>R:u(r)=0\}$. We normalize the
diffusion coefficient to be on unit order, consider the convection vector field
to be on order $r^m$, $m\in R$, pointing either inward $(-)$ or outward $(+)$,
and consider the reaction coefficient $\Lambda$ to be on order $r^{-j}$, $j\in
R$. For both the Neumann boundary case and the Dirichlet boundary case, we show
for which choices of $m$, $(\pm)$ and $j$ a free boundary exists, and when it
exists, we obtain its growth rate in $h$ as a function of $m$, $(\pm)$ and $j$.
These results are then used to study the free boundary in the non-radially
symmetric case.
