On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms

In this paper, we study the elliptic equations -G(alpha)u + c(x) . del(alpha) u = h(x)e(u), x = (x, y) is an element of R-N1 x R-N2 = R-N, where G(alpha) = Delta(alpha) + (1 + alpha)(2)vertical bar x vertical bar(2 alpha)Delta(y), alpha > 0, is the Grushin operator. Here, the advection term c(x) is a smooth, divergence free vector field satisfying certain decay condition and h(x) is a continuous function such that h(x) >= C vertical bar x vertical bar(l), l >= 0, where vertical bar x vertical bar is the Grushin norm of x. We will prove that the equation has no stable solutions provided that N-alpha < 10 + 4l, where N-alpha := N-1 + (1 + alpha)N-2 is the homogeneous dimension of R-N associated to the Grushin operator.