Liouville-type theorem for nonlinear elliptic equations involving p-Laplace-type Grushin operators

In this article, we prove the Liouville-type theorem for stable solutions of weighted p-Laplace-type Grushin equations -divG(a(z)|. Gu| p-2. Gu) = h(z) eu, z = (x,..). RN = RN1 x RN2 (1) and divG(a(z)|. Gu| p-2. Gu) = h(z) u-q, z = (x,..). RN = RN1 x RN2, (2) where p = 2, q > 0 and a(z), h(z). L1 loc(RN) are nonnegative functions satisfying a(z) = C1|| z|| b G and h(z) = C2|| z||.. G as || z|| G = R0 with p -N.. < b <.. + p, R0, Ci(i = 1, 2) are some positive constants. divG(.., g) = SN1 i= 1.... i.. xi + (1 +..)| x|.. SN2.. = 1.. g........,(.., g). C1(RN, RN1 x RN2),. G = (. x, (1 +..)| x|... y),.. = 0, z = (x,..). RN = RN1 x RN2 and || z|| G = (| x| 2(1+..) + |.. | 2) 1 2(1+..). The results hold true for N.. <.. 0(p, b,..) in (1) and q > qc(p, N.., b,..) in (2). Here,.. 0 and qc are new exponents, which are always larger than the classical critical ones and depend on the parameters p, b and... N-gamma = N1 + (1 + gamma) N-2 is the homogeneous dimension of R-N.