#### Vol. 8, No. 7, 2015

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A pointwise inequality for the fourth-order Lane–Emden equation

### Mostafa Fazly, Jun-cheng Wei and Xingwang Xu

Vol. 8 (2015), No. 7, 1541–1563
##### Abstract

We prove the pointwise inequality

where ${c}_{n}:=8∕\left(n\left(n-4\right)\right)$, for positive bounded solutions of the fourth-order Hénon equation, that is,

for some $a\ge 0$ and $p>1$. Motivated by Moser’s proof of Harnack’s inequality as well as Moser iteration-type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient $2∕\left(n-4\right)$ also appears in the fourth-order $Q$-curvature and the Paneitz operator. This, in particular, implies that the scalar curvature of the conformal metric with conformal factor ${u}^{4∕\left(n-4\right)}$ is positive.

##### Keywords
semilinear elliptic equations, a priori pointwise estimate, Moser iteration-type arguments, elliptic regularity
##### Mathematical Subject Classification 2010
Primary: 35B45, 35B50, 35J30, 53C21, 35B08
##### Milestones
Revised: 5 May 2015
Accepted: 29 July 2015
Published: 18 September 2015
##### Authors
 Mostafa Fazly Department of Mathematical and Statistical Sciences CAB 632 Central Academic Building University of Alberta Edmonton, AB T6G 2G1 Canada Department of Mathematics and Computer Science University of Lethbridge Lethbridge, AB T1K 3M4 Canada Jun-cheng Wei Department of Mathematics The University of British Columbia Vancouver, BC V6T 1Z2 Canada Xingwang Xu Department of Mathematics National University of Singapore Block S17 (SOC1) 10 Lower Kent Ridge Road Singapore 119076 Singapore