<p>In this paper, we are concerned about the steady state problem of a degenerate predator-prey model with cross-diffusion. First, some properties of principal eigenvalues are deduced. Next, the stability of the semitrivial steady state solutions are obtained. Finally, existence, uniqueness and stability of positive steady state solutions are derived. The result reveals that the diffusion with spatial degeneracy profoundly influences the structure and stability of semitrivial steady state solutions and the coexistence region, which is a strong contrast to the result with no spatial degeneracy.</p>