Liouville-type theorem for Kirchhoff equations involving Grushin operators

The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations: - M(integral(RN)omega(z)vertical bar del(G)u vertical bar(2) dz)div(G)(omega(z)del(G)u) = f (z)e(u), z = (x, y) is an element of R-N = R-N1 x R-N2 (0.1) and M(integral(RN)omega(z)vertical bar del(G)u vertical bar(2) dz)div(G)(omega(z)del(G)u) = f (z)(u-q), z = (x, y) is an element of R-N = R-N1 x R-N2, (0.2) where M(t) = a + bt(k), t >= 0, with a > 0, b, k >= 0, k = 0 if and only if b = 0. q > 0 and omega(z), f (z) is an element of L-loc(1)(R-N) are nonnegative functions satisfying omega(z) = C-1 parallel to z parallel to(theta)(G) and f(z) >= C-2 parallel to z parallel to(d)(G) as parallel to z parallel to(G) >= R-0 with d > theta - 2, R-0, C-i (i = 1, 2) are some positive constants, here alpha >= 0 and parallel to z parallel to(G) = (vertical bar x vertical bar(2(1+alpha)) + vertical bar y vertical bar(2))(1/2(1+alpha)) is the norm corresponding to the Grushin distance. N-alpha = N-1 + (1 + alpha)N-2 is the homogeneous dimension of R-N. div(G) (resp., del(G)) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, theta, d, and N-alpha, the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.