Eqn. 2 is obtained is valid in Monge gauge and is obtained by INCONSISTENT expansion of the bending energy. The eqn. 2 should contain one extra term: 1/2*c_0^2*(grad u)^2. Please see Gozdz, W.T.; Gompper, G., Europhys Lett. 55(4), pp 587. Inclusion of this extra term would add one extra term to eqn. 6 which would involve c_0*laplace (u). That indicates, that the spontaneous curvature effects the membrane deformation not only via boundary condition but also through bulk term. It's not very clear from the paper, if the author wish to use c_0 = 0.