-
Previous Article
Entropy range problems and actions of locally normal groups
- DCDS Home
- This Issue
-
Next Article
On the Closing Lemma problem for the torus
Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension
| 1. | Department of Mathematics, Wayne State University, Detroit, MI 48202, United States |
| 2. | Department of Mathematics, Information School, Renmin University of China, Beijing 100872, China |
$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$
where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the $L^p$-norm using the method of blow-up analysis:
$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega} e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$
for $0\leq \alpha <\lambda_p(\Omega)$, and the supremum is infinity for all $\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the extremal functions for this inequality when $\alpha$ is sufficiently small.
| [1] |
Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110 |
| [2] |
Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 |
| [3] |
Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212 |
| [4] |
Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 |
| [5] |
Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064 |
| [6] |
Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 |
| [7] |
Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 |
| [8] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
| [9] |
Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047 |
| [10] |
Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941 |
| [11] |
Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 |
| [12] |
Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 |
| [13] |
Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121 |
| [14] |
Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351 |
| [15] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
| [16] |
Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 |
| [17] |
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 |
| [18] |
Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 |
| [19] |
Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 |
| [20] |
Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]