  Building upon the results in [Hinterm\"uller et al., SIAM J. Optim, '15],
generalized Nash equilibrium problems are considered, in which the feasible set
of each player is influenced by the decisions of their competitors. This is
realized via the existence of one (or more) state constraint(s) establishing a
link between the players. Special emphasis is put on the situation of a state
encoded in a possibly non-linear operator equation. First order optimality
conditions under a constraint qualification are derived. Aiming at a
practically meaningful method, an approximation scheme using a penalization
technique leading to a sequence of (Nash) equilibrium problems without
dependence of the constraint set on the other players' strategies is
established. An associated path-following strategy related to a value function
is then proposed. This happens at first on the most abstract level and is
subsequently established to a narrower framework geared to the presence of
partial differential equations in the constraint. Our findings are illustrated
with examples having distributed and boundary controls - both involving
semi-linear elliptic PDEs.
