Classification of the structures of stable radial solutions for semilinear elliptic equations in ${\mathbf{R}}^{\mathbf{N}}$

We study the stability of radial solutions of the semilinear elliptic equation $\mathrm{\Delta }u+f(u)=0$ in ${\mathbf{R}}^{\mathbf{N}}$, where $N\ge 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with respect to the stability of radial solutions, and establish criteria for the existence and nonexistence of stable radial solutions in terms of the limits of $f^{\mathrm{\prime }}(u)F(u)$ as $u\to 0$ or $\mathrm{\infty }$, where $F(u)={\int }_u^{\mathrm{\infty }}1\mathrm{/}f(t)dt$. Furthermore, we show the relation between the existence of singular stable solutions and the solution structure.