  Motivated by its relation to models of flame propagation, we study globally
Lipschitz solutions of $\Delta u=f(u)$ in $\mathbb{R}^n$, where $f$ is smooth,
non-negative, with support in the interval $[0,1]$. In such setting, any
"blow-down" of the solution $u$ will converge to a global solution to the
classical one-phase free boundary problem of Alt-Caffarelli.
  In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study
here the 1D symmetry of solutions $u$ that are energy minimizers. Our main
result establishes that, in dimensions $n<6$, if $u$ is axially symmetric and
stable then it is 1D.
