Large number of fast decay ground states to Matukuma-type equations

In this article we consider the Matukuma type equationequation(∗)Δu+K(|x|)up=0inRN, for positive radially symmetric solutions. When K satisfies some suitable monotonicity assumption, there exists a unique ground state of (∗). In this work we find a large class of K functions for which this monotonicity assumption fails and a large number of bubble-tower ground states exist.