On a Bernoulli-type overdetermined free boundary problem

In this article we study a Bernoulli-type free boundary problem and generalize work of Henrot Shahgholian in [25] to \(\mathcal{A}\)-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the \(p\)-Laplace equation for fixed \(10\) given constant, then there exists unique domain \(\Omega\) with \(K\subset \Omega\) function \(u\) which \(\Omega\setminus K\), has continuous values 1 \(\partial K\) 0 \(\partial\Omega\), such \(|\nabla u|=c\) \Omega\). Moreover, \(\partial\Omega\) \(C^{1,\gamma}\) some \(\gamma>0\), it smooth provided \(\mathcal{A}\) \(\mathbf{R}^n \setminus \{0\}\). We also super level sets \(\{u>t\}\) \(t\in (0,1)\).