We study and obtain formulas for the asymptotic behavior as |x| of C2 solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in n and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as |x|, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results. 1993 Springer-Verlag.
Archive for Rational Mechanics and Analysis