Bifurcation in a variational problem on a surface with a distance constraint

We describe a variational problem on a surface of a Euclidean space under a distance constraint. We provide su cient and necessary conditions for the existence of bifurcation points, generalizing Skrypnik s analog described in [P. Vyridis, Int. J. Nonlinear Anal. Appl. 2 (2011), 1{10]. The problem in local coordinates corresponds to an elliptic boundary value problem.