We consider the eigenvalue problem
$ -\Delta v = \lambda ( c_0 p u^{p-1}_\varepsilon + \varepsilon) v$ in $\Omega,$
$ v = 0$ on $\partial\Omega,$
$ || v ||_{L^\infty(\Omega)} = 1$
where $\Omega \subset R^N (N \ge 5)$ is a smooth bounded domain, $c_0 = N(N-2)$,
$p = (N+2)/(N-2)$ is the critical Sobolev exponent and $\varepsilon >0$ is a small parameter.
Here $u_\varepsilon $ is a positive solution of
$ -\Delta u = c_0 u^p + \varepsilon u $ in $ \Omega,
\quad u|_{\partial \Omega} = 0
$
with the property that
$ \frac{\int_\Omega |\nabla u_\varepsilon |^2 dx}
{( \int_\Omega |u_\varepsilon |^{p+1} dx )^{\frac{2}{p+1}}}
\to S_N$ as $\varepsilon\to 0, $
where $S_N$ is the best constant for the Sobolev inequality.
In this paper, we show several asymptotic estimates for the eigenvalues $\lambda_{i, \varepsilon}$
and corresponding eigenfunctions $v_{i,\varepsilon}$ for $i=1, 2, \cdots, N+1, N+2$.
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