The existence problem of multiple nontrivial solutions for a discrete generalized Emden-Fowler equation which is resonant at zero eigenvalue both at zero and infinity simultaneously was studied.The matrix form and the corresponding energy functional of the above problem were given

then the desired solutions were equivalent to the critical pointsof the energy functional by variational methods.When the nonlinear term met certain assumptions at zero and infinity

it was proved that the problem had at least one and two nontrivial solutions respectively via Morse theory and computations of the critical groups.The conclusion enriches research results of discrete generalized Emden-Fowler equations

which is significant for the further research on other discrete nonlinear problems.