We consider the semilinear elliptic eigenvalue problem $-\Delta u +k(|x|)u^p=\lambda u$ , $u>0$ in $B_R$ , $u=0$ on $\partial B_R$ , where $p>1$ is a constant, $B_R:=\{x\in \text{\mathbfbf {R}}^N:|x |0$ is a parameter. We investigate the global structure of the branch of $(\lambda, u_\lambda)$ of bifurcation diagram from a point of view of $L^2$ -theory. To do this, we establish a precise asymptotic formula for $\lambda=\lambda(\alpha)$ as $\alpha\rightarrow\infty$ , where $\alpha:=\|u_\lambda\|_2$ .