I would be tempted to try to define the sphere in spherical polar co-ordinates, so you have the surface (ie where the radius is one) defined by two varying angles. Then rewrite Del^2 u as its equivalant for the two angles, bang in the boundary condition and see where it takes you. I will see if it works tommorow and learn latex at the same time!

Also if its applied to a sphere ie a 3d shape, i would have thought u=u(u,v,w) thus Del^2 (u) would be " uxx + uyy + uzz = 0 "