  In this paper, first we study carefully the positive solutions to $\Delta
u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact
Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as
Lichnerowicz-type equations, and obtain the gradient estimates of positive
solutions to these equations which do not depend on the bounds of the solutions
and the Laplacian of the distance function on $(M, g)$. Then, we extend our
techniques to a class of more general semilinear elliptic equations $\Delta
u(x)+uh(\ln u)=0$ and obtain some similar results under some suitable analysis
conditions on these equations. Moreover, we also obtain some Liouville-type
theorems for these equations when $Ric(g)\geq 0$ and establish some Harnack
inequalities as consequences.
