Stability of entire solutions to supercritical elliptic problems involving advection

Abstract

We examine the equation given by (1)−Δu+a(x)⋅∇u=upin RN, where p>1 and a(x) is a smooth vector field satisfying some decay conditions. We show that for p<pc, the Joseph–Lundgren exponent, there is no positive stable solution of (1) provided one imposes a smallness condition on a along with a divergence free condition. In the other direction we show that for N≥4 and p>N−1N−3 there exists a positive solution of (1) provided a satisfies a smallness condition. For p>pc we show the existence of a positive stable solution of (1) provided a is divergence free and satisfies a smallness condition.