Abstract

We study the positive solutions to a class of general semilinear elliptic equations $\mathrm{\Delta }u(x)+uh(\ln <!--FUNCTION\; APPLICATION-->u)=0$ defined on a complete Riemannian manifold $(M,g)$ with $\mathrm{Ric}<!--FUNCTION\; APPLICATION-->(g)\ge -Kg$, and obtain Li–Yau-type 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 Liouville-type theorems for these equations when $(M,g)$ is noncompact and $\mathrm{Ric}<!--FUNCTION\; APPLICATION-->(g)\ge 0$ and establish some Harnack inequalities as consequences. As applications of the main theorem, we extend our techniques to the Lichnerowicz-type equations $\mathrm{\Delta }u+{\lambda }_{1}u+{\lambda }_{2}u\ln <!--FUNCTION\; APPLICATION-->u+{\lambda }_3{u}^{b+1}+{\lambda }_4{u}^{p+1}=0$, the Einstein-scalar field Lichnerowicz equations $\mathrm{\Delta }u+{\lambda }_{1}u+{\lambda }_2{u}^{b+1}+{\lambda }_3{u}^{p+1}=0$ with $\dim <!--FUNCTION\; APPLICATION-->(M)\ge 3$ and the two-dimensional Einstein-scalar 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.

Keywords

Mathematical Subject Classification

Milestones

Authors