Ngô Quốc Anh

January 4, 2014

A Picone type identity for bi-Laplacian

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 22:22

Simultaneously, I have recently found the following identity in the same fashion of the Picone identity for \Delta. It says that

\displaystyle \left( \Delta u -\frac uv \Delta v\right)^2-\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2 = |\Delta u|^2 -\Delta \left( \frac {u^2}v\right)\Delta v

for any function v \ne 0. It is worth noticing that the original Picone identity says that

\displaystyle \left| \nabla u - \frac{u}{v}\nabla v\right|^2= \left|\nabla u\right|^2 - \nabla \left( \frac{u^2}{v} \right) \cdot \nabla v \geqslant 0

for any function v \ne 0. It turns out that a few days ago, this identity appeared in a recent notes by Dwivedi  and Tyagi, see Lemma 2.1 from here. The extra term

\displaystyle\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2

naturally appears since it only involves up to third order derivatives. However, to compare this term with 0, we only need to assume that \Delta v has a fixed sign. To see how this identity could be useful, let us consider the following equation

\displaystyle (-\Delta)^2 u +hu= f u ^\frac {n+4}{n-4}, \quad u>0, h<0

naturally arises from prescribing Q-curvature in Riemannian manifolds of dimension n \geqslant 5.

In fact, it was proved by J.C. Wei and X. Xu, here, that solutions of the following equation

\displaystyle (-\Delta)^p u = u ^q \quad \text{ in } \mathbb R^n

satisfy (-\Delta)^iu>0 for any i=\overline {1,p-1}. In particular, when p=2, solutions of the preceding equation fulfill

\Delta u<0 \quad \text{ in }\mathbb R^n.

This property of solutions of this type of equation has been recently improved by W. Cheng and C. Li, see Theorem 2.1 in here, where they replace the equation by the following differential inequality

\displaystyle (-\Delta)^p u \geqslant u ^q \quad \text{ in } \mathbb R^n

On manifolds, I believe that the same situation holds in the presence of the candidate f, i.e. \Delta u<0 at points for which f is positive. If so, we could obtain some non-existence results as I have already done in my previous paper.


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