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Abstract
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We study the positive solutions to a class of general semilinear elliptic equations
defined on a complete
Riemannian manifold
with
,
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
.
We also obtain some Liouville-type theorems for these equations when
is noncompact
and
and establish some Harnack inequalities as consequences. As applications of the
main theorem, we extend our techniques to the Lichnerowicz-type equations
, the Einstein-scalar field
Lichnerowicz equations
with
and the two-dimensional Einstein-scalar field Lichnerowicz equation
, and
obtain some similar gradient estimates and Liouville theorems under some suitable
analysis conditions on these equations.
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Keywords
gradient estimate, Ricci curvature, Liouville theorem,
Harnack inequality, nonlinear elliptic equations
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Mathematical Subject Classification
Primary: 35J15, 53C21
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Milestones
Received: 30 September 2021
Revised: 13 March 2022
Accepted: 26 March 2022
Published: 14 July 2022
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