For parameter dependent nonlinear elliptic boundary value problems a
computer-assisted method enclosing solution branches and bifurcation points
is presented. Starting from finitely many approximate solutions giving rise
to the conjecture that there might be a bifurcation, the method encloses
continuous solution branches, at the same time proving that bifurcation does
indeed take place. It is based on bounds for the defects of the approximate
solutions and on eigenvalue bounds for the linearization at these
approximate solutions.