- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
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 $i=1, 2, \cdots, N+1, N+2$.
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. |
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. |
[1] |
Naoyuki Ishimura, Shin'ya Matsui. On blowing-up solutions of the Blasius equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 985-992. doi: 10.3934/dcds.2003.9.985 |
[2] |
Yessine Dammak. Blowing-up solutions for a supercritical elliptic equation. Communications on Pure and Applied Analysis, 2022, 21 (2) : 625-637. doi: 10.3934/cpaa.2021191 |
[3] |
Mokhtar Kirane, Ahmed Alsaedi, Bashir Ahmad. Blowing-up solutions of differential equations with shifts: A survey. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022100 |
[4] |
Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729 |
[5] |
Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47 |
[6] |
Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure and Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013 |
[7] |
Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929 |
[8] |
Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 |
[9] |
Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 |
[10] |
M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure and Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233 |
[11] |
Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 |
[12] |
Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007 |
[13] |
Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 |
[14] |
Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707 |
[15] |
Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357 |
[16] |
Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 |
[17] |
Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773 |
[18] |
Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567 |
[19] |
Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 |
[20] |
Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]