The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.