    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 & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
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show all references

##### References:
  G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18.  doi: doi:10.1016/0022-1236(91)90099-Q.  Google Scholar  K. Cerqueti, A uniqueness result for a semilinear elliptic equation involving the critical Sobolev exponent in symmetric domains,, Asymptotic Anal., 21 (1999), 99. Google Scholar  M. Grossi and F. Pacella, On an eigenvalue problem related to the critical exponent,, Math. Z., 250 (2005), 225.  doi: doi:10.1007/s00209-004-0755-8.  Google Scholar  Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent,, Ann. Inst. Henri Poincar\'e, 8 (1991), 159. Google Scholar  O. Rey, Proof of two conjectures of H. Brezis and L. A. Peletier,, Manuscripta Math., 65 (1989), 19.  doi: doi:10.1007/BF01168364.  Google Scholar  F. Takahashi, Asymptotic nondegeneracy of least energy solutions to an elliptic problem with critical Sobolev exponent,, Advanced Nonlin. Stud., 8 (2008), 783. Google Scholar
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