The sharpness of some results on stable solutions of in

In this note we give a complete answer to a question raised by Dupaigne and Farina (2009) [8] related to the existence of nonconstant stable solutions of the equation − Δ u = f ( u )  in  R N , where N ≤ 9 and f is a very general non-negative, non-decreasing and convex nonlinearity.