  We consider a free boundary problem for the $p$-Laplace operator which is
related to the so-called Bernoulli free boundary problem. In this formulation,
the classical boundary gradient condition is replaced by a condition on the
distance between two different level surfaces of the solution. For suitable
scalings our model converges to the classical Bernoulli problem; one of the
advantages in this new formulation lies in the simplicity of the arguments,
since one does not need to consider the boundary gradient.
  We shall study this problem in convex and other regimes, and establish
existence and qualitative theory.
