April  2001, 7(2): 275-281. doi: 10.3934/dcds.2001.7.275

A note on a class of higher order conformally covariant equations

1. 

Department of Mathematics, Princeton University and UCLA, United States

2. 

Department of Mathematics, Southwest Missouri State University, United States

Revised  September 2000 Published  January 2001

In this paper, we study the higher order conformally covariant equation

$(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$

for all even dimensions n.
Let

$\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$

We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.

Citation: Sun-Yung Alice Chang, Wenxiong Chen. A note on a class of higher order conformally covariant equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 275-281. doi: 10.3934/dcds.2001.7.275
[1]

Ali Hyder, Juncheng Wei. Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2757-2764. doi: 10.3934/cpaa.2019123

[2]

Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3525-3533. doi: 10.3934/dcdss.2020247

[3]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227

[4]

Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233

[5]

Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1

[6]

Sergi Simon. Linearised higher variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[7]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[8]

Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure and Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811

[9]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013

[10]

Xiaojun Zheng, Zhongdan Huan, Jun Liu. On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2679-2700. doi: 10.3934/cpaa.2022068

[11]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic and Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205

[12]

Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081

[13]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic and Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015

[14]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure and Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163

[15]

Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure and Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519

[16]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589

[17]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239

[18]

Tomás Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, José Valero. Non--autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 415-443. doi: 10.3934/dcds.2008.21.415

[19]

Philip Korman. On uniqueness of positive solutions for a class of semilinear equations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 865-871. doi: 10.3934/dcds.2002.8.865

[20]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control and Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (172)
  • HTML views (0)
  • Cited by (31)

Other articles
by authors

[Back to Top]