  In this article we study a Bernoulli-type free boundary problem and
generalize a work of Henrot and Shahgholian in \cite{HS1} to
$\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose
structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In
particular, we show that if $K$ is a bounded convex set satisfying the interior
ball condition and $c>0$ is a given constant, then there exists a unique convex
domain $\Omega$ with $K\subset \Omega$ and a function $u$ which is
$\mathcal{A}$-harmonic in $\Omega\setminus K$, has continuous boundary values
$1$ on $\partial K$ and $0$ on $\partial\Omega$, such that $|\nabla u|=c$ on
$\partial \Omega$. Moreover, $\partial\Omega$ is $C^{1,\gamma}$ for some
$\gamma>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n
\setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex
for $t\in (0,1)$.
