In this article, we study the weighted Lane-Emden equation
$ \begin{equation*} {\rm div}_{G}\big(\omega_{1}(z)|\nabla_{G}u|^{p-2}\nabla_{G}u\big) = \omega_{2}(z)|u|^{q-1}u, \ z = (x, y)\in \mathbb{R}^{N} = \mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{equation*} $
where $ 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} $.
Citation: Yunfeng Wei, Hongwei Yang, Hongwang Yu. On stable solutions of the weighted Lane-Emden equation involving Grushin operator[J]. AIMS Mathematics, 2021, 6(3): 2623-2635. doi: 10.3934/math.2021159
In this article, we study the weighted Lane-Emden equation
$ \begin{equation*} {\rm div}_{G}\big(\omega_{1}(z)|\nabla_{G}u|^{p-2}\nabla_{G}u\big) = \omega_{2}(z)|u|^{q-1}u, \ z = (x, y)\in \mathbb{R}^{N} = \mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{equation*} $
where $ 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} $.
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Yunfeng Wei, Hongwei Yang, Hongwang Yu. On stable solutions of the weighted Lane-Emden equation involving Grushin operator[J]. AIMS Mathematics, 2021, 6(3): 2623-2635. doi: 10.3934/math.2021159
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