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

In this paper, we are concerned with the double-phase problem involving the Grushin operator in the whole space RN=RN_1N_2 - divG (|∇G u|ᵖ⁻²∇G u + w(z) |∇G u|q-2∇G u) = f(z)|u|ʳ⁻¹u, where ∇G is the Grushin gradient, ΔG is the Grushin operator, q≥ p ≥ 2 , r>q-1 and w, f ∈ L¹loc(RN) 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.