On radial solutions of certain semi-linear elliptic equations

We consider the semi-linear elliptic equation Delta u + f (x, u) + g (vertical bar x vertical bar)x center dot del u = 0, in some exterior region of R-n, n >= 3. It is shown that if f 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. (c) 2005 Elsevier Ltd. All rights reserved.