Analysis & PDE
- Anal. PDE
- Volume 8, Number 7 (2015), 1541-1563.
A pointwise inequality for the fourth-order Lane–Emden equation
We prove the pointwise inequality
where , for positive bounded solutions of the fourth-order Hénon equation, that is,
for some and . 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 also appears in the fourth-order -curvature and the Paneitz operator. This, in particular, implies that the scalar curvature of the conformal metric with conformal factor is positive.
Anal. PDE, Volume 8, Number 7 (2015), 1541-1563.
Received: 9 October 2013
Revised: 5 May 2015
Accepted: 29 July 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35B45: A priori estimates 35B50: Maximum principles 35J30: Higher-order elliptic equations [See also 31A30, 31B30] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 35B08: Entire solutions
Fazly, Mostafa; Wei, Jun-cheng; Xu, Xingwang. A pointwise inequality for the fourth-order Lane–Emden equation. Anal. PDE 8 (2015), no. 7, 1541--1563. doi:10.2140/apde.2015.8.1541. https://projecteuclid.org/euclid.apde/1510843164