Asymptotic behavior of stable radial solutions of semilinear elliptic equations in RN

This paper is devoted to the study of stable nonconstant radial solutions of - Delta u = f(u) in R-N, where f is an element of C-1 (R). We prove that any stable nonconstant bounded radial solution satisfies N > 10 and |u(r) - u(infinity)| >= Mr(-N/2+root N-1+2) for every r >= 1, for certain M > 0, where u(infinity) = lim(r ->infinity) u(r). Moreover, we establish that every stable nonconstant (not necessarily bounded) radial solution satisfies |u(r)| >= Mr(-N/2+root N-1+2) if N not equal 10, and |u(r)| >= M log(r) if N = 10; for r >= r(0), for some M, r(0) > 0. The result is optimal for every N >= 1, but there is a subtle difference between the cases N >= 2 and N = 1. In the first case there are stable radial solutions satisfying lim(r ->infinity) u(r)/r(-N/2+root N-l+2) = 1 if N not equal 10, and lim(r ->infinity) u(r)/log(r) = 1 if N = 10. In the case N = 1 we give a characterization of the stable nonconstant even solutions, which implies lim(r ->infinity) |u(r)|/r(3/2) = +infinity for such functions. This exponent is optimal since, for every s > 3/2, it is possible to find stable even solutions satisfying u(r) = r(S) for every r >= 1. In fact, the techniques we use in both cases are completely different. (C) 2007 Elsevier Masson SAS. All rights reserved.