Liouville-type theorems for the weighted static Choquard equation involving Grushin operator

We are concerned with the weighted static Choquard equation: −divG(ω1(z)|∇Gu|p−2∇Gu)=ω2(z)|u|q−1u∫RNω2(s)|u(s)|q+1‖z−s‖GτdsinRN:=RN1×RN2, where p≥2, q>p−1, ωi(z)∈Lloc1(RN)(i=1,2) are nonnegative functions such that ω1(z)≤C1‖z‖Gθ and ω2(z)≥C2‖z‖Gd for large ‖z‖G with d>θ−p. Here ‖z‖G=(|x|2(1+α)+|y|2)12(1+α),z=(x,y)∈RN1×RN2,α≥0. divG (resp., ∇G) is Grushin divergence (resp., Grushin gradient). We establish various Liouville-type theorems for nontrivial weak solutions belonging to one of the classes: stable solutions and finite Morse index solutions (whether unbounded or sign-changing) under some mild assumptions on q,θ,d,τ and Nα:=N1+(1+α)N2. The main tools we use are energy method and a Pohožaev-type identity.