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An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent

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  • 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 $i=1, 2, \cdots, N+1, N+2$.

    Mathematics Subject Classification: Primary: 35J60, 35J25; Secondary: 35B33.


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