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 Temam (Arch Ration Mech Anal 60(1):51-73, 1975-1976), and Berestycki and Brezis (Nonlinear Anal. 4(3):415-436, 1980): find a function and a constant , satisfying Delta u + g(x, u) = p(x) in D u vertical bar(partial derivative D) = b, integral(partial derivative D)partial derivative u/partial derivative n ds = 0. Here D subset of 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.