Vol. 8, No. 7, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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

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.

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
Received: 9 October 2013
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