This paper is devoted to the study of stable nonconstant radial solutions of − Δ u = f ( u ) in R N , where f ∈ C 1 ( R ) . We prove that any stable nonconstant bounded radial solution satisfies N > 10 and | u ( r ) − u ∞ | ⩾ M r − N / 2 + N − 1 + 2 for every r ⩾ 1 , for certain M > 0 , where u ∞ = lim r → ∞ u ( r ) . Moreover, we establish that every stable nonconstant (not necessarily bounded) radial solution satisfies | u ( r ) | ⩾ M r − N / 2 + N − 1 + 2 if N ≠ 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 → ∞ u ( r ) / r − N / 2 + N − 1 + 2 = 1 if N ≠ 10 , and lim r → ∞ 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 → ∞ | u ( r ) | / r 3 / 2 = + ∞ 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.