In this paper we prove symmetry for solutions to the semi-linear elliptic equation
Δ
u
=
f
(
u
)
in
B
1
,
0
≤
u
>
M
,
in
B
1
,
u
=
M
,
on
∂
B
1
,
\begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*}
where
M
>
0
M>0
is a constant, and
B
1
B_1
is the unit ball. Under certain assumptions on the r.h.s.
f
(
u
)
f (u)
, the
C
1
C^1
-regularity of the free boundary
∂
{
u
>
0
}
\partial \{u>0\}
and a second order asymptotic expansion for
u
u
at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of
C
2
C^2
-regularity of solutions.
Radial symmetry for an elliptic PDE with a free boundary