Abstract: In this paper, the upper bounds of the parameter c for the equation \Delta w = c(w - w^p - 1) on Riemannian manifolds are estimated using the integration by parts method and Liouville's theorem, thereby simplifying the existing computational approaches. The computational methods employed are helpful in providing estimates of the optimal constants in the corresponding Sobolev-type interpolation inequalities and verifying the best constants of the Sobolev inequalities on Riemannian manifolds in the case of spheres.