We study the positive solutions to a class of general semilinear elliptic equations
$\mathrm{\Delta}u(x)+uh(\mathrm{ln}u)=0$ defined on a complete
Riemannian manifold
$(M,g)$
with
$\mathrm{Ric}(g)\ge Kg$,
and obtain Li–Yautype gradient estimates of positive solutions to these equations which
do not depend on the bounds of the solutions or the Laplacian of the distance function
on
$(M,g)$.
We also obtain some Liouvilletype theorems for these equations when
$(M,g)$ is noncompact
and
$\mathrm{Ric}(g)\ge 0$
and establish some Harnack inequalities as consequences. As applications of the
main theorem, we extend our techniques to the Lichnerowicztype equations
$\mathrm{\Delta}u+{\lambda}_{1}u+{\lambda}_{2}u\mathrm{ln}u+{\lambda}_{3}{u}^{b+1}+{\lambda}_{4}{u}^{p+1}=0$, the Einsteinscalar field
Lichnerowicz equations
$\mathrm{\Delta}u+{\lambda}_{1}u+{\lambda}_{2}{u}^{b+1}+{\lambda}_{3}{u}^{p+1}=0$
with
$\mathrm{dim}(M)\ge 3$
and the twodimensional Einsteinscalar field Lichnerowicz equation
$\mathrm{\Delta}u+A{e}^{2u}+B{e}^{2u}+D=0$, and
obtain some similar gradient estimates and Liouville theorems under some suitable
analysis conditions on these equations.
