On global solutions to semilinear elliptic equations related to the one-phase free boundary problem

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.