Singular and Bounded Positive Radial Solutions for Supercritical Elliptic Equations in \(\mathbb {R}^N\)

We study positive radial solutions of elliptic equations \(\Delta u+f(u)=0\) in the entire \(\mathbb {R}^N\) , \(N\ge 3\) , when f has a supercritical growth. When the growth rate is in between the critical Sobolov exponent and the Joseph-Lundgren exponent, we show that the problem has a positive radial singular solution if and only if a set of positive radial bounded solutions is not uniformly bounded. Moreover, we give three examples which exhibit various phenomena depending on the growth rate.