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- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2021
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-021-00425-2
- Publisher site
- See Article on Publisher Site
Abstract
We investigate here the following weighted degenerate elliptic system −Δsu=(1+∥x∥2(s+1))α2(s+1)vp,−Δsv=(1+∥x∥2(s+1))α2(s+1)uθ,u,v>0in RN:=RN1×RN2,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document} $$\begin{aligned} &-\Delta _{s} u =\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{ \frac{\alpha }{2(s+1)}} v^{p}, \quad -\Delta _{s} v= \Big(1+\| \mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha }{2(s+1)}}u^{\theta }, \\ &\quad u,v>0 \quad \mbox{in }\; \mathbb{R}^{N}:=\mathbb{R}^{N_{1}}\times \mathbb{R}^{N_{2}}, \end{aligned}$$ \end{document} where Δs=Δx+|x|2sΔy\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Delta _{s}=\Delta _{x}+|x|^{2s}\Delta _{y}$\end{document}, is the Grushin operator, s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$s$\end{document}, α≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \geq 0$\end{document} and 1<p≤θ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$1< p\leq \theta $\end{document}. Here ∥x∥=(|x|2(s+1)+|y|2)12(s+1)andx:=(x,y)∈RN:=RN1×RN2.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^{2}\Big)^{\frac{1}{2(s+1)}} \; \mbox{and}\quad \mathbf{x}:=(x, y)\in \mathbb{R}^{N}:=\mathbb{R}^{N_{1}} \times \mathbb{R}^{N_{2}}. $$\end{document}In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results (Duong and Phan in J. Math. Anal. Appl. 454(2):785–801, 2017; Hajlaoui et al. in Discrete Contin. Dyn. Syst. 37:265–279, 2017).
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jul 29, 2021
Keywords: Stable solutions; Liouville-type theorem; Weighted Grushin equation; Weighted elliptic system