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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

Futoshi Takahashi 1,

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

Keywords: blowing-up solutions, eigenvalue problem., Critical Sobolev exponent.
Mathematics Subject Classification: Primary: 35J60, 35J25; Secondary: 35B3.
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
References:
[1]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: doi:10.1016/0022-1236(91)90099-Q.  Google Scholar

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Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, 8 (1991), 159-174.  Google Scholar

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O. Rey, Proof of two conjectures of H. Brezis and L. A. Peletier, Manuscripta Math., 65 (1989), 19-37. doi: doi:10.1007/BF01168364.  Google Scholar

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F. Takahashi, Asymptotic nondegeneracy of least energy solutions to an elliptic problem with critical Sobolev exponent, Advanced Nonlin. Stud., 8 (2008), 783-797.  Google Scholar

show all references

References:
[1]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: doi:10.1016/0022-1236(91)90099-Q.  Google Scholar

[2]

K. Cerqueti, A uniqueness result for a semilinear elliptic equation involving the critical Sobolev exponent in symmetric domains, Asymptotic Anal., 21 (1999), 99-115.  Google Scholar

[3]

M. Grossi and F. Pacella, On an eigenvalue problem related to the critical exponent, Math. Z., 250 (2005), 225-256. doi: doi:10.1007/s00209-004-0755-8.  Google Scholar

[4]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, 8 (1991), 159-174.  Google Scholar

[5]

O. Rey, Proof of two conjectures of H. Brezis and L. A. Peletier, Manuscripta Math., 65 (1989), 19-37. doi: doi:10.1007/BF01168364.  Google Scholar

[6]

F. Takahashi, Asymptotic nondegeneracy of least energy solutions to an elliptic problem with critical Sobolev exponent, Advanced Nonlin. Stud., 8 (2008), 783-797.  Google Scholar

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