Advances in Differential Equations

A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system

Liang-Gen Hu

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $\alpha >-4$, $ 0 \le \beta < N-4$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and reveal their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems for stable solutions (whether positive or sign-changing) of the weighted elliptic system in the higher dimension.

Article information

Source
Adv. Differential Equations Volume 22, Number 1/2 (2017), 49-76.

Dates
First available in Project Euclid: 20 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.ade/1484881285

Subjects
Primary: 35B33: Critical exponents 35B45: A priori estimates 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35J60: Nonlinear elliptic equations

Citation

Hu, Liang-Gen. A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system. Adv. Differential Equations 22 (2017), no. 1/2, 49--76. http://projecteuclid.org/euclid.ade/1484881285.


Export citation