Stability of entire solutions to supercritical elliptic problems involving advection

We examine the equation given by equation(1)−Δu+a(x)⋅∇u=upin  RN, where p>1p>1 and a(x)a(x) is a smooth vector field satisfying some decay conditions. We show that for pN−1N−3 there exists a positive solution of (1) provided aa satisfies a smallness condition. For p>pcp>pc we show the existence of a positive stable solution of (1) provided aa is divergence free and satisfies a smallness condition.