Decay rate and radial symmetry of the exponential elliptic equation

Let n⩾3n⩾3, α,β∈Rα,β∈R, and let v   be a solution of Δv+αev+βx⋅∇ev=0Δv+αev+βx⋅∇ev=0 in RnRn, which satisfies the conditions limR→∞1logR∫1Rρ1−n(∫Bρevdx)dρ∈(0,∞) and |x|2ev(x)⩽A1|x|2ev(x)⩽A1 in RnRn. We prove that v(x)log|x|→−2 as |x|→∞|x|→∞ and α>2βα>2β. As a consequence we prove that there exists a constant R0>0R0>0 such that if the solution v(x)v(x) is radially symmetric for |x|