#### Vol. 317, No. 2, 2022

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Gradient estimates and Liouville theorems for Lichnerowicz equations

### Pingliang Huang and Youde Wang

Vol. 317 (2022), No. 2, 363–386
##### Abstract

We study the positive solutions to a class of general semilinear elliptic equations $\mathrm{\Delta }u\left(x\right)+uh\left(\mathrm{ln}u\right)=0$ defined on a complete Riemannian manifold $\left(M,g\right)$ with $\mathrm{Ric}\left(g\right)\ge -Kg$, 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 $\left(M,g\right)$. We also obtain some Liouville-type theorems for these equations when $\left(M,g\right)$ is noncompact and $\mathrm{Ric}\left(g\right)\ge 0$ and establish some Harnack inequalities as consequences. As applications of the main theorem, we extend our techniques to the Lichnerowicz-type equations $\mathrm{\Delta }u+{\lambda }_{1}u+{\lambda }_{2}u\mathrm{ln}u+{\lambda }_{3}{u}^{b+1}+{\lambda }_{4}{u}^{p+1}=0$, the Einstein-scalar field Lichnerowicz equations $\mathrm{\Delta }u+{\lambda }_{1}u+{\lambda }_{2}{u}^{b+1}+{\lambda }_{3}{u}^{p+1}=0$ with $\mathrm{dim}\left(M\right)\ge 3$ 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.

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