Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 7 (2015), 1541-1563.

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

Mostafa Fazly, Jun-cheng Wei, and Xingwang Xu

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Abstract

We prove the pointwise inequality

Δu ( 2 (p + 1) cn)1 2 |x|a2u(p+1)2 + 2 n 4 |u|2 u  in n,

where cn := 8(n(n 4)), for positive bounded solutions of the fourth-order Hénon equation, that is,

Δ2u = |x|aup in n

for some a 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(n 4) 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 u4(n4) is positive.

Article information

Source
Anal. PDE, Volume 8, Number 7 (2015), 1541-1563.

Dates
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
https://projecteuclid.org/euclid.apde/1510843164

Digital Object Identifier
doi:10.2140/apde.2015.8.1541

Mathematical Reviews number (MathSciNet)
MR3399131

Zentralblatt MATH identifier
1328.35035

Subjects
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

Keywords
semilinear elliptic equations a priori pointwise estimate Moser iteration-type arguments elliptic regularity

Citation

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


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