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

Abstract

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} $.

Keywords

• stable weak solutions
• liouville-type theorem
• grushin operator
• lane-emden nonlinearity