Liouville-type theorems for double-phase problems involving the Grushin operator

Abstract

In this paper, we are concerned with the double-phase problem involving the Grushin operator in the whole space ${\mathbb{R}}^N={\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}$

where ${\nabla }_G$ is the Grushin gradient, ${\Delta }_G$ is the Grushin operator, $q\ge p\ge 2$, $r>q-1$ and $w,f\in L_{\mathrm{loc}}^1({\mathbb{R}}^N)$ are two nonnegative functions satisfying some growth conditions at infinity. Our purpose is to establish some Liouville-type theorems for stable weak solutions or for weak solutions which are stable outside a compact set of the equation above.