Comptes Rendus
no. 6
Partial differential equations
Stability for entire radial solutions to the biharmonic equation with negative exponents
Presented by: Haïm Brézis
Xia Huang1; Dong Ye2; Feng Zhou1
1Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China
2IECL, UMR 7502, Département de mathématiques, Université de Lorraine, 3, rue Augustin-Fresnel, 57073 Metz, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 632-636

In this note, we are interested in entire solutions to the semilinear biharmonic equation

Δ2u=−u−p,u>0inRN,
where p>0 and N≥3. In particular, the stability outside a compact set of the entire radial solutions will be completely studied, which resolves the remaining case in [5].

Dans cette note, on s'intéresse aux solutions radiales entières de l'équation semilinéaire biharmonique

Δ2u=−u−p,u>0dansRN,
où p>0 et N≥3. En particulier, on étudie la stabilité en dehors d'un compact des solutions radiales entières, et on résout un cas ouvert dans [5].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.05.001

Xia Huang  1 ; Dong Ye  2 ; Feng Zhou  1

1 Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China
2 IECL, UMR 7502, Département de mathématiques, Université de Lorraine, 3, rue Augustin-Fresnel, 57073 Metz, France 
@article{CRMATH_2018__356_6_632_0,
     author = {Xia Huang and Dong Ye and Feng Zhou},
     title = {Stability for entire radial solutions to the biharmonic equation with negative exponents},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {632--636},
     year = {2018},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     doi = {10.1016/j.crma.2018.05.001},
     language = {en},
}
TY  - JOUR
AU  - Xia Huang
AU  - Dong Ye
AU  - Feng Zhou
TI  - Stability for entire radial solutions to the biharmonic equation with negative exponents
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 632
EP  - 636
VL  - 356
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2018.05.001
LA  - en
ID  - CRMATH_2018__356_6_632_0
ER  - 
%0 Journal Article
%A Xia Huang
%A Dong Ye
%A Feng Zhou
%T Stability for entire radial solutions to the biharmonic equation with negative exponents
%J Comptes Rendus. Mathématique
%D 2018
%P 632-636
%V 356
%N 6
%I Elsevier
%R 10.1016/j.crma.2018.05.001
%G en
%F CRMATH_2018__356_6_632_0
Xia Huang; Dong Ye; Feng Zhou. Stability for entire radial solutions to the biharmonic equation with negative exponents. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 632-636. doi: 10.1016/j.crma.2018.05.001

[1] P. Caldiroli; R. Musina Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., Volume 45 (2012) no. 1–2, pp. 147-164

[2] J. Dávila; I. Flores; I. Guerra Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., Volume 348 (2010) no. 1, pp. 143-193

[3] I. Guerra A note on nonlinear biharmonic equations with negative exponents, J. Differ. Equ., Volume 253 (2012) no. 11, pp. 3147-3157

[4] T. Kusano; M. Naita; C.A. Swanson Radial entire solutions of even order semilinear elliptic equations, Can. J. Math., Volume 40 (1988), pp. 1281-1300

[5] B.S. Lai The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents, Proc. R. Soc. Edinb., Sect. A, Volume 146 (2016) no. 1, pp. 195-212

[6] B.S. Lai; D. Ye Remarks on entire solutions for two fourth-order elliptic problems, Proc. Edinb. Math. Soc., Volume 59 (2016) no. 3, pp. 777-786

[7] P.J. McKenna; W. Reichel Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differ. Equ., Volume 2003 (2003) no. 37 (13 p)

[8] G. Warnault Liouville theorems for stable radial solutions for the biharmonic operator, Asymptot. Anal., Volume 69 (2010), pp. 87-98

Cited by Sources:

Comments - Policy