On stable solutions of the weighted Lane-Emden equation involving Grushin operator

Abstract

<abstract><p>In this article, we study the weighted Lane-Emden equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>where $ N = N_{1}+N_{2}\geq2, $ $ p\geq2 $ and $ q &gt; 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 &gt; \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} $.</p></abstract>