Structure of the sets of regular and singular radial solutions for a semilinear elliptic equation

This paper is concerned with the structure of the set of radially symmetric solutions for the equationΔu+f(u)=0onRn⧹{0},with n>2n>2. Here the nonlinear term ff is assumed to be a smooth function of uu that is positive for u>0u>0 and is equal to 0 for u⩽0u⩽0. Then any radial solution u=u(r),r=|x|, of the equation is shown to be classified into one of several types according to its behavior as r→0r→0 and r→∞r→∞. Under the assumption that ff is supercritical for small u>0u>0 and is subcritical for large u>0u>0, we clarify the entire structure of the set of solutions of various types. The Pohozaev identity plays a crucial role in the investigation of the structure.