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Let $u$ be non-negative smooth subharmonic function on a complete Riemannian manifold $M$. Take $v=u^{p/2}$ with $p>1$. Then in this paper

Yau, Shing-Tung, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25, 659-670 (1976); erratum ibid. 31, 607 (1982). ZBL0335.53041,

(more precisely formula (2.19) on the 5th page), it is given that

$$v\Delta v=\frac{p}{2}vu^{p/2-1}\Delta u+\frac{p}{2}(\frac{p}{2}-1)vu^{p/2-2}|du|^2.$$

I can not prove the above equality. Please help me by giving some intermediate steps.
Thank you.

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  • $\begingroup$ Maybe the chain rule can be useful. $\endgroup$ – paf Jun 29 '18 at 20:23

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