Gradient estimates for the 𝑝-Laplacian Lichnerowicz equation on smooth metric measure spaces

<p>In this paper, we consider the weighted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Laplacian Lichnerowicz equation <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="white up pointing triangle u plus c u Superscript sigma Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">△<!-- △ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>σ<!-- σ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \triangle _{p,f} u+cu^{\sigma }=0 \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> on smooth metric measure spaces, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than-or-equal-to 0 comma p greater-than 1 comma"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">c\geq 0, p>1,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma less-than-or-equal-to p minus 1"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma \leq p-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real constants. A local gradient estimate for positive solutions to this equation is derived, and as applications, we give a corresponding Liouville property and Harnack inequality.</p>

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