    August  2011, 4(4): 907-922. doi: 10.3934/dcdss.2011.4.907

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

 1 Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Received  September 2009 Revised  January 2010 Published  November 2010

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 $i=1, 2, \cdots, N+1, N+2$.

Citation: Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
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##### References:
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