Department of Informatics and Telecommunications, Kalamata Technological Educational Institute, Branch of Sparta, 23100 Sparta, Greece.
We describe a variational problem on a surface under a constraint of geometrical character. Necessary and sufficient conditions for the existence of bifurcation points are provided. In local coordinates the problem corresponds to a quasilinear elliptic boundary value problem. The problem can be considered as a physical model for several applications referring to continuum medium and membranes.