The author investigates the bifurcation behaviour of solutions of variational problems with smallest eigenvalue of the second variation zero and whose third variation in an eigenfunction direction which does not vanish. Then there are bifurcations, that means that there are two branches of critical points of the variational integral if the boundary varies a little bit, one branch minimizing and the other one not. It was shown in the two-dimensional situation by \textit{L. Lichtenstein} [``Untersuchungen über zweidimensionale reguläre Variationsprobleme'', Monatsh. Math. 28, 3-51 (1917)]. The author treats these kinds of variational problems on manifolds and gives applications to minimal surfaces.