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 Δu(x) + uh(ln u) = 0 defined on a complete Riemannian manifold (M,g) with Ric(g) 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 Ric(g) 0 and establish some Harnack inequalities as consequences. As applications of the main theorem, we extend our techniques to the Lichnerowicz-type equations Δu + λ1u + λ2uln u + λ3ub+1 + λ4up+1 = 0, the Einstein-scalar field Lichnerowicz equations Δu + λ1u + λ2ub+1 + λ3up+1 = 0 with dim (M) 3 and the two-dimensional Einstein-scalar field Lichnerowicz equation Δu + Ae2u + Be2u + 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
Authors
Pingliang Huang
Department of Mathematics
Shanghai University
Shanghai
China
Youde Wang
School of Mathematics and Information Sciences
Guangzhou University
Guangzhou
China
Hua Loo-Keng Key Laboratory of Mathematics
Institute of Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing
China
School of Mathematical Sciences
University of Chinese Academy of Sciences
Beijing
China