  Minimizing the so-called "Dirichlet energy" with respect to the domain under
a volume constraint is a standard problem in shape optimization which is now
well understood. This article is devoted to a prototypal non-linear version of
the problem, where one aims at minimizing a Dirichlet-type energy involving the
solution to a semilinear elliptic PDE with respect to the domain, under a
volume constraint. One of the main differences with the standard version of
this problem rests upon the fact that the criterion to minimize does not write
as the minimum of an energy, and thus most of the usual tools to analyze this
problem cannot be used. By using a relaxed version of this problem, we first
prove the existence of optimal shapes under several assumptions on the problem
parameters. We then analyze the stability of the ball, expected to be a good
candidate for solving the shape optimization problem, when the coefficients of
the involved PDE are radially symmetric.
