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Singular backward self-similar solutions of a semilinear parabolic equation
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 |
$ -\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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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
F. Takahashi, Asymptotic nondegeneracy of least energy solutions to an elliptic problem with critical Sobolev exponent, Advanced Nonlin. Stud., 8 (2008), 783-797. |
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