Gradient estimates and Liouville theorems for Lichnerowicz equations

Pingliang Huang and Youde Wang

Vol. 317 (2022), No. 2, 363–386

DOI: 10.2140/pjm.2022.317.363

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

gradient estimate, Ricci curvature, Liouville theorem, Harnack inequality, nonlinear elliptic equations

Mathematical Subject Classification

Primary: 35J15, 53C21

Milestones

Received: 30 September 2021

Revised: 13 March 2022

Accepted: 26 March 2022

Published: 14 July 2022