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} - \operatorname{div}_{G} (|\nabla_{G} u|^{p-2}\nabla_{G} u + w(z) |\nabla_{G} u|^{q-2}\nabla_{G} u) = f(z)|u|^{r-1}u, where \nabla_{G} is the Grushin gradient, \Delta_{G} is the Grushin operator, q\geq p \geq 2 , r>q-1 and w, f \in L^{1}_{\mathrm{loc}}(\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.