On stable solutions of the weighted Lane-Emden equation involving Grushin operator
In this article, we study the weighted Lane-Emden equationwhere $ N = N_{1}+N_{2}\geq2, $ $ p\geq2 $ and $ q > p-1 $, while $ \omega_{i}(z)\in L^{1}_{\rm loc}(\mathbb{R}^{N})\setminus\{0\}(i = 1, 2) $ are nonnegative functions satisfying $ \omega_{1}(z)\leq C\|z\|_{G}^{\theta} $ and $ \omega_...
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doaj-a7d31a41adfc46c2af75bdee9d07d91d2021-01-12T01:37:37ZengAIMS PressAIMS Mathematics2473-69882021-01-01632623263510.3934/math.2021159On stable solutions of the weighted Lane-Emden equation involving Grushin operatorYunfeng Wei0Hongwei Yang1Hongwang Yu21. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, P. R. China2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, P. R. China1. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, P. R. ChinaIn this article, we study the weighted Lane-Emden equationwhere $ N = N_{1}+N_{2}\geq2, $ $ p\geq2 $ and $ q > p-1 $, while $ \omega_{i}(z)\in L^{1}_{\rm loc}(\mathbb{R}^{N})\setminus\{0\}(i = 1, 2) $ are nonnegative functions satisfying $ \omega_{1}(z)\leq C\|z\|_{G}^{\theta} $ and $ \omega_{2}(z)\geq C'\|z\|_{G}^{d} $ for large $ \|z\|_{G} $ with $ d > \theta-p. $ Here $ \alpha\geq0 $ and $ \|z\|_{G} = (|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}. $ $ \rm div_{G} $ (resp., $ \nabla_{G} $) is Grushin divergence (resp., Grushin gradient). We prove that stable weak solutions to the equation must be zero under various assumptions on $ d, \theta, p, q $ and $ N_{\alpha} = N_{1}+(1+\alpha)N_{2} $.http://awstest.aimspress.com/article/doi/10.3934/math.2021159?viewType=HTMLstable weak solutionsliouville-type theoremgrushin operatorlane-emden nonlinearity |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yunfeng Wei Hongwei Yang Hongwang Yu |
spellingShingle |
Yunfeng Wei Hongwei Yang Hongwang Yu On stable solutions of the weighted Lane-Emden equation involving Grushin operator AIMS Mathematics stable weak solutions liouville-type theorem grushin operator lane-emden nonlinearity |
author_facet |
Yunfeng Wei Hongwei Yang Hongwang Yu |
author_sort |
Yunfeng Wei |
title |
On stable solutions of the weighted Lane-Emden equation involving Grushin operator |
title_short |
On stable solutions of the weighted Lane-Emden equation involving Grushin operator |
title_full |
On stable solutions of the weighted Lane-Emden equation involving Grushin operator |
title_fullStr |
On stable solutions of the weighted Lane-Emden equation involving Grushin operator |
title_full_unstemmed |
On stable solutions of the weighted Lane-Emden equation involving Grushin operator |
title_sort |
on stable solutions of the weighted lane-emden equation involving grushin operator |
publisher |
AIMS Press |
series |
AIMS Mathematics |
issn |
2473-6988 |
publishDate |
2021-01-01 |
description |
In this article, we study the weighted Lane-Emden equationwhere $ N = N_{1}+N_{2}\geq2, $ $ p\geq2 $ and $ q > p-1 $, while $ \omega_{i}(z)\in L^{1}_{\rm loc}(\mathbb{R}^{N})\setminus\{0\}(i = 1, 2) $ are nonnegative functions satisfying $ \omega_{1}(z)\leq C\|z\|_{G}^{\theta} $ and $ \omega_{2}(z)\geq C'\|z\|_{G}^{d} $ for large $ \|z\|_{G} $ with $ d > \theta-p. $ Here $ \alpha\geq0 $ and $ \|z\|_{G} = (|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}. $ $ \rm div_{G} $ (resp., $ \nabla_{G} $) is Grushin divergence (resp., Grushin gradient). We prove that stable weak solutions to the equation must be zero under various assumptions on $ d, \theta, p, q $ and $ N_{\alpha} = N_{1}+(1+\alpha)N_{2} $. |
topic |
stable weak solutions liouville-type theorem grushin operator lane-emden nonlinearity |
url |
http://awstest.aimspress.com/article/doi/10.3934/math.2021159?viewType=HTML |
work_keys_str_mv |
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