A theoretical model for describing the adhesion of lipid vesicle with free edges is developed. For adhesion in contact potential or in finite-range potential, the total energy functional is defined as the sum of elastic free energy, the surface energy, the line tension energy and the contact potential or the long-ranged potential. The equilibrium differential equation and boundary conditions for opening-up lipid vesicles are derived through minimizing the total energy functional. Numerical solutions to these equations are obtained under the axial symmetric condition. These numerical solutions can be used to qualitatively explain the influence of the substrate on the open-up lipid vesicles.