  We study the existence and multiplicity of solutions and the global solution
curve of the following free boundary value problem, arising in plasma physics,
see R. Temam [18], and H. Berestycki and H. Brezis [3]: find a function $u(x)$
and a constant $b$, satisfying \[ \Delta u+g(x,u)=p(x) \;\; \mbox{in $D$} \] \[
  u \,| \, _{\partial D}=b, \;\;\;\; \int_{\partial D} \frac{\partial
u}{\partial n} \, ds=0 \,. \] Here $D \subset R^n$, is a bounded domain, with a
smooth boundary. This problem can be seen as a PDE generalization of the
periodic problem for one-dimensional pendulum-like equations. We use
continuation techniques. Our approach is suitable for numerical computations.
