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July/August 2012 Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems
Quoc Hung Phan
Adv. Differential Equations 17(7/8): 605-634 (July/August 2012).

Abstract

We consider the Hardy-Hénon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e., the nonexistence of positive solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a natural conjecture that this property should be true if and only if $(N+a)/(p+1)+(N+b)/(q+1)>(N-2)$. In this paper, we prove the conjecture for dimension $N=3$ in the case of bounded solutions and in dimensions $N\le 4$ when $a,b\le 0$, among other partial nonexistence results. As far as we know, this is the first optimal Liouville-type result for the Hardy-H\'enon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.

Citation

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Quoc Hung Phan. "Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems." Adv. Differential Equations 17 (7/8) 605 - 634, July/August 2012.

Information

Published: July/August 2012
First available in Project Euclid: 17 December 2012

zbMATH: 1262.35108
MathSciNet: MR2963798

Subjects:
Primary: 35B33, 35B40, 35B45, 35B53, 35J60

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.17 • No. 7/8 • July/August 2012
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