On radial solutions of certain semi-linear elliptic equations

We consider the semi-linear elliptic equation Δu+f(x,u)+g(|x|)x·∇u=0Δu+f(x,u)+g(|x|)x·∇u=0, in some exterior region of Rn,n⩾3Rn,n⩾3. It is shown that if ff depends radially on its first argument and is nonincreasing in its second, boundary conditions force the unique solution to be radial. Under different conditions, we prove the existence of a positive radial asymptotic solution to the same equation.